  We show how one can use Hermite-Pad\'{e} approximation and little $q$-Jacobi
polynomials to construct rational approximants for $\zeta_q(2)$. These numbers
are $q$-analogues of the well known $\zeta(2)$. Here $q=\frac{1}{p}$, with $p$
an integer greater than one. These approximants are good enough to show the
irrationality of $\zeta_q(2)$ and they allow us to calculate an upper bound for
its measure of irrationality: $\mu(\zeta_q(2))\leq 10\pi^2/(5\pi^2-24) \approx
3.8936$. This is sharper than the upper bound given by Zudilin (\textit{On the
irrationality measure for a $q$-analogue of $\zeta(2)$}, Mat. Sb. \textbf{193}
(2002), no. 8, 49--70).
