Does there exist $a \in \mathbb{Q}$ such that $a^2 - a + 1$ is a square?
$$a^2-a+1=(a-\frac{1}{2})^2+\frac{3}{4}$$
So what you ask are there any rational numbers $u=a-\frac{1}{2}$ and $v$ such that,
$$u^2+\frac{3}{4}=v^2$$
$$v^2-u^2=\frac{3}{4}$$
$$(v-u)(v+u)=\frac{3}{4}$$
Let $x$ and $y$ be two real numbers that multiply to $\frac{3}{4}$, with the condition that $\frac{x+y}{2}$ and $\frac{y-x}{2}$ are both rational. Note that this condition equates to the condition that both $x$ and $y$ are rational.
Solving the system,
$$v-u=x$$
$$v+u=y$$
Yields the rational solutions $v=\frac{x+y}{2}$ and $u=\frac{y-x}{2}$. Which implies $a=\frac{y-x+1}{2}$.
If you restrict $v$ to be an integer. Then it must be that,
$$x+y=2v \in \mathbb{Z}$$
$$xy=\frac{3}{4}$$
Eliminating $y$ from this system gives,
$$x(2v-x)=\frac{3}{4}$$
Or,
$$4x^2-8vx+3=0$$
By the rational root theorem, the only rational solutions to the above equation are,
$$x=\pm \frac{3}{2}, \pm \frac{3}{4}, \pm 3, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 1$$
The only solutions which correspond to integer values of $v$ are $x=\pm \frac{3}{2}, \pm \frac{1}{2}$. These give $v=\pm 1$. So we have that,
$$x+y=\pm 2$$
$$xy=\frac{3}{4}$$
Solving for $x$ and $y$ leads to the conclusion that the only possible integer solutions for $a$ is $a=0$ or $a=1$.
For the record, integers $m,n$ such that $m^2-mn+n^2$ is a perfect square (integer) do exist, for instance $m=15$, $n=8$, for which $m^2-mn+n^2=169=13^2$. This is just to say that the answer to at least one unambiguous form of the question is affirmative.
An integer square multiple of one of your rational triples becomes an integer triple.
integer solutions ( with $a,b > 0$) of $a^2 - ab + b^2 = c^2$ are called Eisenstein triples, see https://en.wikipedia.org/wiki/Integer_triangle#Integer_triangles_with_a_60.C2.B0_angle_.28angles_in_arithmetic_progression.29
You would then get $$ \left( \frac{a}{b} \right)^2 - \left( \frac{a}{b} \right) + 1 = \left( \frac{c}{b} \right)^2 $$
I found an alternative parametrization: we can take $\gcd(u,v) = 1,$ also $u \neq v \pmod 3,$ then $0 < u < v,$ then take $$ a = u^2 + 2uv, \; \; b = 2uv+ v^2, \; \; c = u^2 + uv + v^2 $$
If you want $ab$ negative, you can replace $(a,b,c)$ by $(-a, -a+b,c),$ among other choices. Or, just allow $uv < 0.$ Put another way, if you begin with $a < 0, b > 0,$ replace $(a,b,c)$ by $(-a, -a+b,c)$ to get all positive. It all ends the same. If anyone can figure out the entire first line, please let me know, lyrics not posted anywhere easy to find. At about 27 seconds in, he says "A strongman from a carnival was in my dreams last night. He had an GIBBERISH and he rode upon a bike." GIBBERISH is about seconds 31-32.
a^2 - a b + b^2 = c^2
c a b c u v c
7 5 8 7 1 2 7 = 7
13 7 15 13 1 3 13 = 13
19 16 21 19 2 3 19 = 19
31 11 35 31 1 5 31 = 31
37 33 40 37 3 4 37 = 37
43 13 48 43 1 6 43 = 43
49 39 55 49 3 5 49 = 7^2
61 56 65 61 4 5 61 = 61
67 32 77 67 2 7 67 = 67
73 17 80 73 1 8 73 = 73
79 51 91 79 3 7 79 = 79
91 19 99 91 1 9 91 = 7 * 13
91 85 96 91 5 6 91 = 7 * 13
97 57 112 97 3 8 97 = 97
103 40 117 103 2 9 103 = 103
109 95 119 109 5 7 109 = 109
127 120 133 127 6 7 127 = 127
133 23 143 133 1 11 133 = 7 * 19
133 88 153 133 4 9 133 = 7 * 19
139 69 160 139 3 10 139 = 139
151 115 171 151 5 9 151 = 151
157 25 168 157 1 12 157 = 157
163 75 187 163 3 11 163 = 163
169 161 176 169 7 8 169 = 13^2
181 104 209 181 4 11 181 = 181
193 175 207 193 7 9 193 = 193
199 56 221 199 2 13 199 = 199
211 29 224 211 1 14 211 = 211
217 208 225 217 8 9 217 = 7 * 31
217 87 247 217 3 13 217 = 7 * 31
223 168 253 223 6 11 223 = 223
229 145 264 229 5 12 229 = 229
241 31 255 241 1 15 241 = 241
247 203 275 247 7 11 247 = 13 * 19
247 93 280 247 3 14 247 = 13 * 19
259 155 299 259 5 13 259 = 7 * 37
259 64 285 259 2 15 259 = 7 * 37
271 261 280 271 9 10 271 = 271
277 217 312 277 7 12 277 = 277
283 192 325 283 6 13 283 = 283
301 136 345 301 4 15 301 = 7 * 43
301 279 319 301 9 11 301 = 7 * 43
307 35 323 307 1 17 307 = 307
313 105 352 313 3 16 313 = 313
331 320 341 331 10 11 331 = 331
337 272 377 337 8 13 337 = 337
343 37 360 343 1 18 343 = 7^3
349 111 391 349 3 17 349 = 349
361 185 416 361 5 16 361 = 19^2
367 315 403 367 9 13 367 = 367
373 152 425 373 4 17 373 = 373
379 259 435 379 7 15 379 = 379
397 385 408 397 11 12 397 = 397
403 333 448 403 9 14 403 = 13 * 31
403 80 437 403 2 19 403 = 13 * 31
409 304 465 409 8 15 409 = 409
421 41 440 421 1 20 421 = 421
427 123 475 427 3 19 427 = 7 * 61
427 240 493 427 6 17 427 = 7 * 61
433 407 455 433 11 13 433 = 433
439 205 504 439 5 18 439 = 439
457 287 527 457 7 17 457 = 457
463 43 483 463 1 21 463 = 463
469 129 520 469 3 20 469 = 7 * 67
469 456 481 469 12 13 469 = 7 * 67
481 215 551 481 5 19 481 = 13 * 37
481 369 544 481 9 16 481 = 13 * 37
487 88 525 487 2 21 487 = 487
499 301 576 499 7 18 499 = 499
511 264 589 511 6 19 511 = 7 * 73
511 451 555 511 11 15 511 = 7 * 73
523 387 595 523 9 17 523 = 523
541 184 609 541 4 21 541 = 541
547 533 560 547 13 14 547 = 547
553 473 608 553 11 16 553 = 7 * 79
553 47 575 553 1 23 553 = 7 * 79
559 141 616 559 3 22 559 = 13 * 43
559 440 629 559 10 17 559 = 13 * 43
571 235 651 571 5 21 571 = 571
577 368 665 577 8 19 577 = 577
589 329 680 589 7 20 589 = 19 * 31
589 559 615 589 13 15 589 = 19 * 31
601 49 624 601 1 24 601 = 601
607 147 667 607 3 23 607 = 607
613 423 703 613 9 19 613 = 613
619 245 704 619 5 22 619 = 619
631 616 645 631 14 15 631 = 631
637 200 713 637 4 23 637 = 7^2 * 13
637 552 697 637 12 17 637 = 7^2 * 13
643 517 720 643 11 18 643 = 643
661 441 760 661 9 20 661 = 661
673 400 777 673 8 21 673 = 673
679 104 725 679 2 25 679 = 7 * 97
679 611 731 679 13 17 679 = 7 * 97
691 539 779 691 11 19 691 = 691
703 312 805 703 6 23 703 = 19 * 37
703 53 728 703 1 26 703 = 19 * 37
709 159 775 709 3 25 709 = 709
721 265 816 721 5 24 721 = 7 * 103
721 705 736 721 15 16 721 = 7 * 103
727 637 792 727 13 18 727 = 727
733 600 817 733 12 19 733 = 733
739 371 851 739 7 23 739 = 739
751 520 861 751 10 21 751 = 751
757 55 783 757 1 27 757 = 757
763 165 832 763 3 26 763 = 7 * 109
763 477 880 763 9 22 763 = 7 * 109
769 735 799 769 15 17 769 = 769
787 112 837 787 2 27 787 = 787
793 385 912 793 7 24 793 = 13 * 61
793 583 903 793 11 21 793 = 13 * 61
811 336 925 811 6 25 811 = 811
817 495 943 817 9 23 817 = 19 * 43
817 800 833 817 16 17 817 = 19 * 43
823 728 893 823 14 19 823 = 823
829 689 920 829 13 20 829 = 829
853 232 945 853 4 27 853 = 853
859 560 989 859 10 23 859 = 859
871 59 899 871 1 29 871 = 13 * 67
871 795 931 871 15 19 871 = 13 * 67
877 177 952 877 3 28 877 = 877
883 715 987 883 13 21 883 = 883
889 295 999 889 5 27 889 = 7 * 127
889 464 1025 889 8 25 889 = 7 * 127
907 413 1040 907 7 26 907 = 907
919 901 936 919 17 18 919 = 919
931 531 1075 931 9 25 931 = 7^2 * 19
931 61 960 931 1 30 931 = 7^2 * 19
937 183 1015 937 3 29 937 = 937
949 305 1064 949 5 28 949 = 13 * 73
949 696 1081 949 12 23 949 = 13 * 73
961 649 1104 961 11 24 961 = 31^2
967 427 1107 967 7 27 967 = 967
973 248 1073 973 4 29 973 = 7 * 139
973 935 1007 973 17 19 973 = 7 * 139
991 549 1144 991 9 26 991 = 991
997 767 1127 997 13 23 997 = 997
1009 496 1161 1009 8 27 1009 = 1009
1021 671 1175 1021 11 25 1021 = 1021
1027 1008 1045 1027 18 19 1027 = 13 * 79
1027 128 1085 1027 2 31 1027 = 13 * 79
1033 928 1113 1033 16 21 1033 = 1033
1039 885 1144 1039 15 22 1039 = 1039
1051 384 1189 1051 6 29 1051 = 1051
1057 65 1088 1057 1 32 1057 = 7 * 151
1057 793 1200 1057 13 24 1057 = 7 * 151
1063 195 1147 1063 3 31 1063 = 1063
1069 744 1225 1069 12 25 1069 = 1069
1087 1003 1155 1087 17 21 1087 = 1087
1093 455 1247 1093 7 29 1093 = 1093
1099 640 1269 1099 10 27 1099 = 7 * 157
1099 915 1219 1099 15 23 1099 = 7 * 157
1117 585 1288 1117 9 28 1117 = 1117
1123 67 1155 1123 1 33 1123 = 1123
1129 201 1216 1129 3 32 1129 = 1129
1141 1121 1160 1141 19 20 1141 = 7 * 163
1141 335 1271 1141 5 31 1141 = 7 * 163
1147 1037 1232 1147 17 22 1147 = 31 * 37
1147 715 1323 1147 11 27 1147 = 31 * 37
1153 992 1265 1153 16 23 1153 = 1153
1159 136 1221 1159 2 33 1159 = 19 * 61
1159 469 1320 1159 7 30 1159 = 19 * 61
1171 896 1325 1171 14 25 1171 = 1171
1183 408 1333 1183 6 31 1183 = 7 * 13^2
1183 603 1363 1183 9 29 1183 = 7 * 13^2
1201 1159 1239 1201 19 21 1201 = 1201
1213 737 1400 1213 11 28 1213 = 1213
1231 680 1421 1231 10 29 1231 = 1231
1237 280 1353 1237 4 33 1237 = 1237
1249 871 1431 1249 13 27 1249 = 1249
1261 1240 1281 1261 20 21 1261 = 13 * 97
1261 71 1295 1261 1 35 1261 = 13 * 97
1267 1152 1357 1267 18 23 1267 = 7 * 181
1267 213 1360 1267 3 34 1267 = 7 * 181
1273 1105 1392 1273 17 24 1273 = 19 * 67
1273 560 1457 1273 8 31 1273 = 19 * 67
1279 355 1419 1279 5 33 1279 = 1279
1291 1005 1456 1291 15 26 1291 = 1291
1297 497 1472 1297 7 32 1297 = 1297
1303 952 1485 1303 14 27 1303 = 1303
1321 639 1519 1321 9 31 1321 = 1321
1327 1235 1403 1327 19 23 1327 = 1327
1333 73 1368 1333 1 36 1333 = 31 * 43
1333 840 1537 1333 12 29 1333 = 31 * 43
1339 1139 1475 1339 17 25 1339 = 13 * 103
1339 219 1435 1339 3 35 1339 = 13 * 103
1351 365 1496 1351 5 34 1351 = 7 * 193
1351 781 1560 1351 11 30 1351 = 7 * 193
1369 511 1551 1369 7 33 1369 = 37^2
1381 296 1505 1381 4 35 1381 = 1381
1387 1365 1408 1387 21 22 1387 = 19 * 73
1387 923 1595 1387 13 29 1387 = 19 * 73
1393 1273 1488 1393 19 24 1393 = 7 * 199
1393 657 1600 1393 9 32 1393 = 7 * 199
1399 1224 1525 1399 18 25 1399 = 1399
1417 1120 1593 1417 16 27 1417 = 13 * 109
1417 592 1617 1417 8 33 1417 = 13 * 109
1423 803 1643 1423 11 31 1423 = 1423
1429 1065 1624 1429 15 28 1429 = 1429
1447 152 1517 1447 2 37 1447 = 1447
1453 1407 1495 1453 21 23 1453 = 1453
1459 949 1680 1459 13 30 1459 = 1459
1471 456 1645 1471 6 35 1471 = 1471
1477 1207 1647 1477 17 27 1477 = 7 * 211
1477 888 1705 1477 12 31 1477 = 7 * 211
1483 77 1520 1483 1 38 1483 = 1483
1489 231 1591 1489 3 37 1489 = 1489
1501 1095 1711 1501 15 29 1501 = 19 * 79
1501 385 1656 1501 5 36 1501 = 19 * 79
1519 1496 1541 1519 22 23 1519 = 7^2 * 31
1519 760 1749 1519 10 33 1519 = 7^2 * 31
1531 1349 1664 1531 19 26 1531 = 1531
1543 693 1768 1543 9 34 1543 = 1543
1549 1241 1736 1549 17 28 1549 = 1549
1561 1184 1769 1561 16 29 1561 = 7 * 223
1561 79 1599 1561 1 39 1561 = 7 * 223
1567 237 1672 1567 3 38 1567 = 1567
1579 395 1739 1579 5 37 1579 = 1579
1591 1064 1829 1591 14 31 1591 = 37 * 43
1591 1491 1675 1591 21 25 1591 = 37 * 43
1597 553 1800 1597 7 36 1597 = 1597
1603 1387 1755 1603 19 27 1603 = 7 * 229
1603 160 1677 1603 2 39 1603 = 7 * 229
1609 1001 1856 1609 13 32 1609 = 1609
1621 711 1855 1621 9 35 1621 = 1621
1627 480 1813 1627 6 37 1627 = 1627
1651 1155 1891 1651 15 31 1651 = 13 * 127
1651 869 1904 1651 11 34 1651 = 13 * 127
1657 1633 1680 1657 23 24 1657 = 1657
1663 1533 1768 1663 21 26 1663 = 1663
1669 1480 1809 1669 20 27 1669 = 1669
1687 1027 1947 1687 13 33 1687 = 7 * 241
1687 1368 1885 1687 18 29 1687 = 7 * 241
1693 328 1833 1693 4 39 1693 = 1693
1699 1309 1920 1699 17 30 1699 = 1699
1723 83 1763 1723 1 41 1723 = 1723
1729 1185 1984 1729 15 32 1729 = 7 * 13 * 19
1729 1679 1775 1729 23 25 1729 = 7 * 13 * 19
1729 249 1840 1729 3 40 1729 = 7 * 13 * 19
1729 656 1961 1729 8 37 1729 = 7 * 13 * 19
c a b c u v c
a^2 - a b + b^2 = c^2