  Orthogonal polynomial solutions of an admissible potentially self-adjoint
linear second-order partial $q$-difference equation of the hypergeometric type
in two variables on $q$-linear lattices are analyzed. A $q$-Pearson's system
for the orthogonality weight function, as well as for the difference
derivatives of the solutions are presented, giving rise to a solution of the
$q$-difference equation under study in terms of a Rodrigues-type formula. The
monic orthogonal polynomial solutions are treated in detail, giving explicit
formulae for the matrices in the corresponding recurrence relations they
satisfy. Lewanowicz and Wo\'zny [S. Lewanowicz, P. Wo\'zny, J. Comput. Appl.
Math. 233 (2010) 1554--1561] have recently introduced a (non-monic) bivariate
extension of big $q$-Jacobi polynomials together with a partial $q$-difference
equation of the hypergeometric type that governs them. This equation is
analyzed in the last section: we provide two more orthogonal polynomial
solutions, namely, a second non-monic solution from the Rodrigues'
representation, and the monic solution both from the recurrence relation that
govern them and also explicitly given in terms of generalized bivariate basic
hypergeometric series. Limit relations as $q \uparrow 1$ for the partial
$q$-difference equation and for the all three $q$-orthogonal polynomial
solutions are also presented.
