  We introduce, characterise and provide a combinatorial interpretation for the
so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key
role in the (reciprocal) expansion of any power of a second order
$q$-differential operator having the $q$-classical polynomials as
eigenfunctions in terms of other even order operators, which we explicitly
construct in this work. The results here obtained can be viewed as the
$q$-version of those given by Everitt {\it et al.} and by the first author,
whilst the combinatorics of this new set of numbers is a $q$-version of the
Jacobi-Stirling numbers given by Gelineau and the second author.
