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

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