  The eigenvalue probability density function of the Gaussian unitary ensemble
permits a $q$-extension related to the discrete $q$-Hermite weight and
corresponding $q$-orthogonal polynomials. A combinatorial counting method is
used to specify a positive sum formula for the spectral moments of this model.
The leading two terms of the scaled $1/N^2$ genus-type expansion of the moments
are evaluated explicitly in terms of the incomplete beta function. Knowledge of
these functional forms allows for the smoothed leading eigenvalue density and
its first correction to be determined analytically.
