In this paper, we propose a method for solving initial-boundary value problems (IBVPs) for hyperbolic integro-differential systems (HIDSs) with functional terms. The equations under consideration involve derivatives with respect to a spatial variable with a generalized piecewise constant argument (GPCA), as well as integral operators acting on the time variable. For a qualitative analysis of such problems, we construct auxiliary problems for first-order integro-differential systems with distributed parameters and additional integral constraints. Using the parameterization of the Dzhumabaev method, the auxiliary problem is reduced to a family of parameterized integral problems. Under explicit invertibility conditions for certain matrices, we establish theorems on the existence and uniqueness of solutions for this family of problems. Furthermore, we develop a constructive iterative algorithm to obtain solutions to the auxiliary problem and prove its convergence. The results extend the theory of HIDEs and open up prospects for applications in models with discontinuities, impulse effects, and integral memory terms.