Define the principal congruence subgroup of level $ n $, $ \Gamma(n) $, to be the subgroup of $ PSL_2(\mathbb{Z}) $ which is congruent to the identity mod $ n $. $ \Gamma(n) $ is discrete infinite, and torsion free.
A (Fuchsian) surface group is also a discrete infinite and torsion free group of 2 by 2 matrices.
Are there any $ n $ for which $ \Gamma(n) $ is a surface group (isomorphic to the fundamental group of a surface)?
No, $PSL(2,\mathbb Z)$ has a free subgroup $F$ of finite index. If $\Gamma(n)$ was a surface group $G$, then $G$ would have a free subgroup $G\cap F$ of finite index. However every finite index subgroup of a surface group is a surface group itself possibly of different genus. Of course I assume that by "surface group" you mean the fundamental group of a closed surface.