We consider a quarter-symmetric metric connection in a Kenmotsu manifold. We investigate the curvature tensor and the Ricci tensor of a Kenmotsu manifold with respect to the quarter-symmetric metric connection. We show that the scalar curvature of an $n$-dimensional locally symmetric Kenmotsu manifold with respect to the quarter-symmetric metric connection is equal to $n(1-n)$. Furthermore, we obtain the non-existence of generalized recurrent, $\phi$-recurrent and pseudosymmetric Kenmotsu manifolds with respect to quarter-symmetric metric connection.