In this paper we study a class of reflected backward stochastic differential equations (RBSDE) driven by the compensated random measure associated to a given pure jump Markov process X on a general state space U. The "reflection" keeps the solution above a given càdlàg process. We prove the uniqueness and existence both by a combination of the Snell envelope theory and fixed point argument. We apply these results to represent probabilitically the value function of some quasi-variational inequalities associated to the Markov process X.