  Let $\Lambda$ be a maximal $\mathbb{F}_q[T]$-order in a division quaternion
algebra over $\mathbb{F}_q(T)$ which is split at the place $\infty$. The
present article gives an algorithm to compute a fundamental domain for the
action of the group of units $\Lambda^*$ on the Bruhat-Tits tree $\mathcal{T}$
associated to $PGL_2(\mathbb{F}_q((1/T)))$. This action is a function field
analog of the action of a co-compact Fuchsian group on the upper half plane.
The algorithm also yields an explicit presentation of the group $\Lambda^*$ in
terms of generators and relations. Moreover we determine an upper bound for its
running time using that $\Lambda^*\backslash\mathcal{T}$ is {\em almost}
Ramanujan.
