Narayana numbers that are products of two Fibonacci numbers

Let $\{N_m{\}}_{m\ge 0}$ be the Narayana’s cows sequence given by $N_0=0$, $N_1=1=N_2=1$ and $$N_{m+3}=N_{m+2}+N_m,\text{ for }m\ge 0$$ and let $\{F_n{\}}_{n\ge 0}$ be the Fibonacci sequence. In this paper we solve explicitely the Diophantine equation $$N_m=F_n{F}_k,$$ in positive unknowns $m,n$ and $k$. That is, we find the non-zero narayana numbers that are products of two Fibonacci numbers.