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Let $\Lambda$ be a maximal $𝔽{}_q\left[T\right]$-order in a division quaternion algebra over $𝔽{}_q\left(T\right)$ 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 $𝒯$ associated to ${\mathrm{PGL}}_2\left(𝔽{}_q\left((1/T)\right)\right)$. 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...