This paper is in continuation of the work performed by Kajla et al. (Applied Mathematics and Computation 2016; 275 : 372–385.) wherein the authors introduced a bivariate extension of q-Bernstein–Schurer–Durrmeyer operators and studied the rate of convergence with the aid of the Lipschitz class function and the modulus of continuity. Here, we estimate the rate of convergence of these operators by means of Peetre's K-functional. Then, the associated generalized Boolean sum operator of the q-Bernstein–Schurer–Durrmeyer type is defined and discussed. The smoothness properties of these operators are improved with the help of mixed K-functional. Furthermore, we show the convergence of the bivariate Durrmeyer-type operators and the associated generalized Boolean sum operators to certain functions by illustrative graphics using Maple algorithm. Copyright © 2017 John Wiley & Sons, Ltd. Copyright © 2017 John Wiley & Sons, Ltd.