In this paper, we consider the Dirichlet problem for a class of Hessian quotient equations involving a gradient term on the right hand sides on Riemannian manifolds. Under the assumption of an admissible subsolution, we solve the existence and the uniqueness for the Dirichlet problem on compact Riemannian manifold, based on the a priori estimates for the solutions to the Hessian quotient type equations. Compared with the classical results for Hessian type equations, our results do not depend on the convexity assumption for the right hand side of the equation.