The function for Fibonacci numbers in Mathematica is called Fibonacci:
I'm not sure how much functions like FullSimplfiy are able to reason about expressions with Fibonacci in them, but the identitiy you wrote doesn't seem to hold. Compare:
Table[Fibonacci[-1 + n] + Fibonacci[n], {n, 1, 10}]
Table[2*Fibonacci[-3 + n], {n, 1, 10}]
