The [wikipedia article][1] is quite enlightening on the Pisano period, but from the algorithmic viewpoint, it shows that you only need to compute the period for $n$ a prime power $p^k,$, and in that case it divides either $p^{k-1}(p-1)$ or $p^{k-1}2(p+1).$ For your range of $n$ brute force will tell you the period quickly, by computing the multiplicative order of $\begin{pmatrix}0&1\\1&1\end{pmatrix}$ modulo $n.$


  [1]: https://en.wikipedia.org/wiki/Pisano_period