  Let $p$ be a prime. In this paper, we give a complete classification of
self-reciprocal polynomials arising from Fibonacci polynomials over
$\mathbb{Z}$ and $\mathbb{Z}_p$, where $p=2$ and $p>5$. We also present some
partial results when $p=3, 5$. We also compute the first and second moments of
Fibonacci polynomials $f_{n}(x)$ over finite fields, which give necessary
conditions for Fibonacci polynomials to be permutation polynomials over finite
fields.
