In the paper: "CONSTRUCTING PAIRS OF PANTS" by Ara Basmajian (Source: http://www.acadsci.fi/mathematica/Vol15/vol15pp065-074.pdf), the author defined:
A torsion free Fuchsian group $G$ is said to be a pair of pants if $H^2/G$ is topologically a sphere with 3 holes. The pair of paints is tight if one of the holes is really a puncture (that is, an open neighborhood of it is hyperbolically a punctured disc).
I knew that a pair of pants with geodesic boundary cannot be a quotient space of $H^2$ by some Fuchsian group because their universal cover is a proper subset of $H^2$. So I guess that with the above definition, we can construct the only pair of pants with 3 cusps. If my guess is wrong, could someone please give me some counterexamples of Fuchsian groups so that the quotient space is a pair of pants with geodesic boundary. Thanks in advance!
The key concept that you are still missing, and which helps tie together all the different contexts, is that a geometric pair of pants can have one of four possibilities: 3 cusps; 2 cusps and 1 geodesic boundary; 1 cusp and 2 geodesic boundaries; or 3 geodesic boundaries. Furthermore, whichever of these is appropriate can be "constructed" from the Fuchsian group $G$.
The point is that the "construction" is not just as simple as taking the quotient of the whole of $H^2$ with respect to the $G$-action, as your question pre-supposes. Instead, the correct construction is to take the quotient of the convex hull of the limit set with respect to the $G$-action.
For example, if the group $G$ has no parabolic elements then the quotient of the convex hull of the limit set with respect to the $G$-action will be a geometric pair of pants with 3 geodesic boundary components. More generally, the number of cusps of the corresponding geometric pair of pants equals the number of conjugacy classes in $G$ of maximal infinite cyclic parabolic subgroups.