In the present paper, we study a large-time behavior of solutions to a quasi-linear second-order hyperbolic system which describes a motion of viscoelastic materials. The system has dissipative properties consisting of a memory term and a damping term. It is proved that the solution exists globally in time in the Sobolev space, provided that the initial data are sufficiently small. Moreover, we show that the solution converges to zero as time tends to infinity. The crucial point of the proof is to derive uniform a priori estimates of solutions by using an energy method.