So, at various places on Wikipedia (like here), its claimed that the $(2,3,7)$ triangle group can be obtained as quotient of the modular group. How is this done?
Treating the integers as mod $7$ will obviously create a finite group (which is apparently isomorphic to the $(2,3,7)$ triangle group, but on the Klein Quartic instead of the entire hyperbolic plane). Here is saying that, treating the elements of the modular group as fractional linear transformations, you can just wrap the space it is acting on mod $7$ in the $x$ axis. It appears that you still have tons of triangles with infinite order vertices though this way.
It seems like you could get the group by modding by the normal closure of $\{T^7\}$, where $T^7$ is the act of rotating about an ideal vertice $7$ times. This may just lead to the Klein Quartic though again, I'm not sure.
Is there some way that $(2,3,7)$ can be obtained as a quotient of the modular group?
The modular group can be generated by the transformations $s\colon z\mapsto -1/z$ and $r\colon z\mapsto -1/z-1$, which obey the relations $s^2=r^3=1$. To get the $(2,3,7)$ triangle group, we need the additional relation $(sr)^7=1$, so we need to take quotient by $z\mapsto z/(1+7z)$.