I'm not certain there is a "real" relation, but there does seem to be a coincidental relationship, particularly if you look at a piano keyboard.
There are 13 chromatic notes from C to C. Of these 5 are black keys (C#, D#, F#, G# A#) and 8 are white keys. Further, the black keys are grouped 2 and 3, because there are 2 white keys (E, F) between (C#, D#) and (F#, G# A#), but only 1 white key between black keys generally.
So you have a manifestation of the numbers 2, 3, 5, 8, and 13, which does bear a superficial relationship to the Fibonacci sequence. I don't see how you extend this, and as I noted, this might just be an interesting coincidence.