It is impossible for $(x-1)^2+x^2+(x+1)^2$ to be a perfect square

As you stated, you let the $3$ consecutive numbers be $x-1$, $x$, and $x+1$. This will give you a sum of their squares to be $3x^2 + 2$. Consider any integer $n$ and $r = 0,1$ or $2$. Then $(3n+r)^2 = 9n^2 + 6nr + r^2$, so the possible remainders when divided by $3$ are just $r^2$, i.e., $0$, $1$, plus $4$ which has a remainder of $1$ also. Thus, all perfect squares have a remainder of either $0$ or $1$ when divided by $3$, but this sum has a remainder of $2$. Thus, it cannot be a perfect square.

In general, you should try to handle these types of questions by checking the remainders (sometimes called congruences in higher math) of various small integers to see if you can find any particular pattern, such as determine anything which doesn't fit.