In this work, we study an initial boundary value problem for a quasi-linear parabolic system with a weak viscoelastic term of the form \begin{equation*} \begin{aligned} A(t) |u_{t}|^{m - 2} u_{t} +\Delta^{2}u - {\alpha}(t) &\int_{0}^{t}g( t - s) \Delta^{2}( s )ds = 0, \qquad x\in \Omega, t \geq 0, \\ u(x,t) = \frac {\partial u} {\partial \nu}(x,t)&= 0,\qquad x\in \partial \Omega,\ t \geq 0,\\ u( x , 0) = u_{0}&(x),\qquad x \in \Omega,\\ \end{aligned} \end{equation*} where $m\geq 2$, $A(t)$ a bounded and positive definite matrix is considered in a bounded domain $\Omega$ in $R^{n}$. We establish a general decay result, which depends on the behavior of both $\alpha$ and $g$, by using the perturbed energy functional technique.