We study certain curvature properties of Kenmotsu manifolds with respect to the quarter-symmetric metric connection. First we consider Ricci semisymmetric Kenmotsu manifolds with respect to the quarter-symmetric metric connection. Next, we study $\xi$-conformally flat and $\xi$-concircularly flat Kenmotsu manifolds with respect to the quarter-symmetric metric connection. Moreover, we study Kenmotsu manifolds satisfying the condition $\tilde Z(\xi,Y)\cdot\tilde S=0$, where $\tilde Z$ and $\tilde S$ are the concircular curvature tensor and Ricci tensor respectively with respect to the quarter-symmetric metric connection. Then, we prove the non-existence of $\xi$-projectively flat and pseudo Riccisymmetric Kenmotsu manifolds with respect to the quarter-symmetric metric connection. Finally, we construct an example of a $5$-dimensional Kenmotsu manifold admitting a quarter-symmetric metric connection for illustration.