The $n=3$'rd Catalan number (A000108) is $1,1,2,5$ : $\frac{\binom{2n}{n}}{n+1}=\frac{\binom{6}{3}}{4}=\frac{20}{4}=$ 5.

The $n=4$'th Fibonacci number (A000045) is $1,1,2,3,5,...$ : 5.

Q. Which other Fibonacci numbers (besides $\{1,2,5\}$) are also Catalan numbers?

There seems to be no other "small" solutions, at least up to Fibonacci/Catalan numbers around $10^{60}$.