In this paper, we established the sharp Chen-type inequalities for $QR$-submanifolds immersed in quaternionic space forms endowed with a quarter-symmetric metric connection (QSMC). By extending Chen's $\delta$-invariant framework to both the invariant and anti-invariant distributions of a $QR$-submanifold, we obtained optimal upper bounds for the invariants $\delta (\mathcal{D})$ and $\delta ({\mathcal{D}}^{\perp })$. The resulting inequalities explicitly capture the influence of the quarter-symmetric metric connection through its structural parameters and associated tensor fields, thereby providing a unified generalization of the classical Levi-Civita and semi-symmetric settings. Furthermore, we completely characterized the equality cases, showing that the bounds are attained precisely when the submanifold is mixed geodesic, and the invariant distribution is totally umbilical under specific constraints on the second fundamental form. As direct consequences, previously known inequalities for semi-symmetric metric and nonmetric connections are recovered. These results furnish a new rigidity phenomena for extremal $QR$-submanifolds and deepen the understanding of curvature invariants in quaternionic geometry.