Irrationality and irrationality measures of a $q$-analogue of $\zeta(2)$

We consider a $q$-analogue of $\zeta(s)=\sum_{k=1}^\infty\frac1{k^s}$, defined as $\zeta_q(s)=\sum_{k=1}^\infty\frac{k^{s-1}q^k}{1-q^k}$. We construct an Hermite-Padé approximation problem for $\zeta_q(2)$ and solve it using little $q$-Legendre polynomials. Careful consideration of the quality of the obtained approximations, including an application of the $q$-Mellin transform, yields the irrationality and an improved upper bound for the irrationality measure of $\zeta_q(2)$, with $q=1/p$, where $p$ is an integer larger than 1.