On the irrationality measure for a $q$-analogue of $\zeta(2)$

A Liouville-type estimate is proved for the irrationality measure of the quantities
$$ \zeta_q(2) =\sum_{n=1}^\infty\frac{q^n}{(1-q^n)^2} $$
with $q^{-1}\in\mathbb Z\setminus\{0,\pm1\}$. The proof is based on the application of a $q$-analogue of the arithmetic method developed by Chudnovsky, Rukhadze, and Hata and of the transformation group for hypergeometric series–the group-structure approach introduced by Rhin and Viola.