A $q$-analogue of the Koecher-Leshchiner generating function of odd zeta values

In the 1980s, Koecher and, independently, Leshchiner found an elegant formula for the generating function of odd zeta values. In this short note, we derive a $q$-analogue of this formula, which provides a $q$-version of the accelerated series for $\zeta (3)$ used by Apéry in his famous proof of irrationality.