Certainly not. Ogg proved in "Real points on Shimura curves" that there are only finitely many Shimura curves (of the form $X^D_0(N)$) which are hyperelliptic. If there is a prime divisor $q$ of $D$ such that $q\equiv 1\bmod 12$ then for all $N$ coprime to $D$, $X^D_0(N) \to X^D_0(1)$ is a connected finite \'etale cover.

Now take $D = 26$. $X^{26}_0(1)$ is of genus 2 and hence is hyperelliptic. Take $N$ large enough and you're done.

I also guess there are easier examples but this was the first one to come to me.