  A semi-infinite weighted Hankel matrix with entries defined in terms of basic
hypergeometric series is explicitly diagonalized as an operator on
$\ell^{2}(\mathbb{N}_{0})$. The approach uses the fact that the operator
commutes with a diagonalizable Jacobi operator corresponding to
Al-Salam-Chihara orthogonal polynomials. Yet another weighted Hankel matrix,
which commutes with a Jacobi operator associated with the continuous
$q$-Laguerre polynomials, is diagonalized. As an application, several new
integral formulas for selected quantum orthogonal polynomials are deduced. In
addition, an open research problem concerning a quantum Hilbert matrix is also
mentioned.
