B&acaron;rbosu and Muraru (2015) introduced the bivariate generalization of the $q$-Bernstein-Schurer-Stancu operators and constructed a GBS operator of $q$-Bernstein-Schurer-Stancu type. The concern of this paper is to obtain the rate of convergence in terms of the partial and complete modulus of continuity and the degree of approximation by means of Lipschitz-type class for the bivariate operators. In the last section we estimate the degree of approximation by means of Lipschitz class function and the rate of convergence with the help of mixed modulus of smoothness for the GBS operator of $q$-Bernstein-Schurer-Stancu type. Furthermore, we show comparisons by some illustrative graphics in Maple for the convergence of the operators to some functions.