Differences of squares of upper-triangular $2\times 2$ integer matrices

We consider the problem of characterizing upper-triangular matrices $M=\left(\begin{array}{cc}p& r\\ 0& q\end{array}\right)\in M_2(\mathbb{Z})$ which can be represented in the form $A^2-B^2$ with upper-triangular integer matrices $A$ and $B$ and give a complete criterion in terms of representations of $p$ and $q$ as differences of two squares and an additional divisibility condition on $r$. Also, we give a complete classification of representable matrices in terms of congruence conditions on $p$, $q$, and $r$.