Let $a,b$ be positive integers with $x^2-ay^2-bz^2+abv^2=0$ having only the zero solution over $\mathbb Z$ and consider the Fuchsian group \begin{equation*} \Gamma=\left\{\begin{bmatrix} k+\sqrt{a}l & m+\sqrt{a}n \\ b(m-\sqrt{a}n) & k-\sqrt{a}l \\ \end{bmatrix} \colon k,l,m,n\in\mathbb Z, k^2-al^2-bm^2+abn^2=1\right\}. \end{equation*} I would like to know a reference where the abelianization of $\Gamma$ is described. Can one find an explicit non-zero group homomorphism $\Gamma\to\mathbb Z$?