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 $\Gamma\to\mathbb Z$?