On a Variant of Pillai's problem involving convergent denominators of quadratic irrationals

Let $(q_{\alpha ,n}{)}_{n\ge 0}$ be the sequence of convergent denominators to the simple continued fraction expansion of $\alpha$. For certain specific choices of $\alpha$, this sequence is a Lehmer sequence. In this paper, we show that there are only finitely many integers $c$ such that the equation $q_{\alpha ,n}-q_{\beta ,m}=c$ has at least two distinct solutions $(n,m)$, where $\alpha ,\beta$ are quadratic irrationals with $\mathbb{Q}(\alpha )\mathrm{\ne }\mathbb{Q}(\beta )$. In specific instances, we solve the equation $q_{\alpha ,n}-q_{\beta ,m}=c$ completely and explicitly list all solutions.