In this paper, we consider the solvability of a system of constrained two-sided coupled generalized Sylvester quaternion matrix equations. Some necessary and sufficient conditions for the existence of solutions to this system are derived in terms of the ranks and the generalized inverses of the coefficient matrices. Moreover, we find an expression of the general solution to this system when the solvability conditions are satisfied. Finally, a numerical example is presented to illustrate our results. (C) 2019 Elsevier Ltd. All rights reserved.