Build an ASCII Fibonacci Clock

Python 2, 194 182 bytes

from random import*
h=m=H,M=input()
while[h,m]!=[H,M/5]:
 h=m=0;s=[]
 for n in 1,1,2,3,5:c=randint(0,3);h+=c%2*n;m+=c/2*n;s=zip(*(["WRGB"[c]*n]*n+s)[::-1])
for L in s:print"".join(L)

The algorithm is just rejection sampling, so it keeps generating clocks until it gets one that's right. The clock is built by starting with nothing, then doing "add a square above and rotate clockwise" 5 times.

Takes two comma-separated integers via STDIN.

>>> ================================ RESTART ================================
>>> 
7,17
BBBWWWWW
BBRWWWWW
RRRWWWWW
RRRWWWWW
RRRWWWWW
>>> ================================ RESTART ================================
>>> 
7,17
GGBRRRRR
GGRRRRRR
WWWRRRRR
WWWRRRRR
WWWRRRRR

CJam, 61 bytes

l~5/]:A{;L[TT][XXYZ5]{4mr_2bW%Mf*@.+\Ps=M*aM*@+W%z\}fMA=!}gN*

Takes two space-separated integers via STDIN, and uses 3.14 instead of WRGB respectively. Try it online.

Here is the "sane" RGBW version for a few extra bytes:

l~5/]:A{;L[TT][XXYZ5]{4mr_2bW%Mf*@.+\"WRGB"=M*aM*@+W%z\}fMA=!}gN*

Explanation

The algorithm is the same as my Python answer — rejection sampling by generating clocks until we get one that's correct.

l~5/]:A            Read input and make array [<hours> <minutes>/5]
{...}g             Do...

  ;                  Pop the only element on the stack
  L                  Push empty array, which will become our clock
  [TT]               Push [0 0] for [h m], to keep track of our sample
  [XXYZ5]{...}fI     For I in [1 1 2 3 5]...
    4mr                Push random number from [0 1 2 3]
    _2bW%              Copy and get reversed base 2 rep for one of [0] [1] [0 1] [1 1]
    If*                Multiply bit(s) by I
    @.+                Add element-wise to [h m] array
    \Ps=               Index the random number into stringified pi for one of "3.14"
    I*aI*              Make into I by I square
    @+W%z\             Add above clock and rotate clockwise

  A=!              ... while the resulting clock is incorrect
N*                 Riffle clock with newlines

Python 2, 421 bytes

Ugh, I'm sure this can be golfed more.

from itertools import*
from random import*
f,r=[1,1,2,3,5],range
c={_:[x for x in chain(*[combinations(f,i)for i in r(6)])if sum(x)==_]for _ in r(13)}
k=[[2,1,4],[2,0,4]]+[[3,4]]*3
def b(h,m):
 o=['W']*5;m/=5;h,m=choice(c[h]),choice(c[m])
 l=dict(zip(zip('WWR',[m,h,m]),'GRB'))
 for x in h,m:
    d={1:[0,1],2:[2],3:[3],5:[4]}
    for _ in x:j=d[_].pop();o[j]=l[o[j],x]
 print'\n'.join([''.join(o[i]*f[i]for i in _)for _ in k])

Test case:

>>> b(7,20)
WWBRRRRR
WWRRRRRR
GGGRRRRR
GGGRRRRR
GGGRRRRR
>>> b(7,20)
RRBWWWWW
RRRWWWWW
BBBWWWWW
BBBWWWWW
BBBWWWWW