  The 1970s conjecture of Lov\'asz and Plummer that the number of perfect
matchings in any $3$-regular graph is exponential in the number of vertices was
proved in 2011 by Esperet, Kardo\v{s}, King, Kr\'al', and Norine. We give the
exact formula for the number of perfect matchings in two families of
$3$-regular graphs. In the graph consisting of a $2n$-cycle with diametric
chords (also known as the M\"obius ladder $M_n$ and a Harary graph) and in the
cartesian product of the cycle $C_n$ with an edge (called the cycle prism), the
number of matchings is the sum of the Fibonacci numbers $F_{n-1}$ and
$F_{n+1}$, plus two more for the M\"obius ladder when $n$ is odd and for the
cycle prism when $n$ is even.
