Let $Q$ be a finite acyclic quiver, let $J$ be an ideal of $kQ$ generated by all arrows in $Q$, and let $A$ be a finite-dimensional $k$ -algebra. The category of all finite-dimensional representations of $\left( Q,\,{{J}^{2}} \right)$ over $A$ is denoted by $\text{rep}\left( Q,\,{{J}^{2}},\,A \right)$ . In this paper, we introduce the category $\text{exa}\left( Q,{{J}^{2}},A \right),$ which is a subcategory of $\text{rep}\left( Q,\,{{J}^{2}},\,A \right)$ of all exact representations. The main result of this paper explicitly describes the Gorenstein-projective representations in $\text{rep}\left( Q,\,{{J}^{2}},\,A \right)$, via the exact representations plus an extra condition. As a corollary, $A$ is a self-injective algebra if and only if the Gorenstein-projective representations are exactly the exact representations of $\left( Q,\,{{J}^{2}} \right)$ over $A$.