  The generalisation of continuous orthogonal polynomial ensembles from random
matrix theory to the $q$-lattice setting is considered. We take up the task of
initiating a systematic study of the corresponding moments of the density from
two complementary viewpoints. The first requires knowledge of the ensemble
average with respect to a general Schur polynomial, from which the spectral
moments follow as a corollary. In the case of little $q$-Laguerre weight, a
particular ${}_3 \phi_2$ basic hypergeometric polynomial is used to express
density moments. The second approach is to study the $q$-Laplace transform of
the un-normalised measure. Using integrability properties associated with the
$q$-Pearson equation for the $q$-classical weights, a fourth order
$q$-difference equation is obtained, generalising a result of Ledoux in the
continuous classical cases.
