In this paper, we delve into the study of quarter-symmetric metric connection $(QSMC)$ on Sasaki-Kenmotsu manifolds. Our investigation encompasses the analysis of the Riemann curvature tensor, Ricci tensor and associated properties within the context of the $QSMC$. Additionally, we establish the non-existence of recurrent, A-recurrent and pseudo-symmetric Sasaki-Kenmotsu manifolds under the $QSMC$. Moreover, we explore conformally flat and $\rho_1$-conformally flat Sasaki-Kenmotsu manifolds with respect to the $QSMC$, presenting several intriguing results. The primary focus of this paper lies in elucidating the implementation of the $QSMC$ on Sasaki-Kenmotsu manifolds.