Does there exist Fuchsian groups, which is not conjugated in $SL(2, \mathbb{R})$ to a subgroup of $SL(2, \mathbb{Z})$, but still contains a congruence subgroup?

How about the normalizer of $\Gamma_0(N)$ in $PSL_2(\mathbb{R})$ for $N > 1$, which is the subgroup generated by $\Gamma_0(N)$ and $\begin{pmatrix} 0 & 1 \\ N & 0 \end{pmatrix}$? (I wouldn't call these "noncongruence subgroups" as you do in your title: that technical term is usually parsed as "(noncongruence) subgroup". This isn't a subgroup of $PSL_2(\mathbb{Z})$ at all, but it certainly contains a principal congruence subgroup; perhaps we should call it a "congruence nonsubgroup"?) 

