Let $K = \mathbb{F}_q(T)$ be the rational function field over the field
$\mathbb{F}_q$ of $q$ elements. Let $\Lambda$ be a maximal order in a division
quaternion algebra $D$ over $K$ which is split at the place $\infty = 1/T$. Let
$K_\infty$ denote the completion of $K$ at $\infty$. Then the group of units
$\Gamma := \Lambda^\star$ acts cocompactly on the Bruhat-Tits tree
$\mathcal{T}$ associated to $PGL_2(K_\infty)$. In this arcticle, we present an
algorithm for computing a fundamental domain for the action of $\Gamma$ on
$\mathcal{T}$.