Presentations of PSL(2, Z/p^n)

A method to do this for the group $\textrm{PSL}_2 (\mathbb{F}_{p^n})$ can be found in the papers by Glover and Sjerve:

Representing $PSl_2(p)$ on a Riemann surface of least genus, L'Enseignement Mathématique 31 (1985)

The genus of $PSl_2(q)$, Journal für die reine und angewandte Mathematik 380 (1987).

I usually use Sunday's presentation: see MR0311782. His T has order 2 but your S will be what he denotes ST.

The group $PSL_2(\mathbb{Z}/p^n)$ is the automorphisms group of the $(p+1)$ regular tree of depth $n$, where at level $m$ of the tree you have the points of $\mathbb{P}(\mathbb{Z}/p^m)$. The main benefit of this view, is that you can understand the relations at each level, and then move inductively to the next one.