In order to describe quantum properties of open quantum system, it is necessary to extend the existing quantum Markov chains. In this paper, Exogenous quantum Markov chains is introduced through building Exogenous quantum operator logic. For this new type of quantum Markov chain, the paper focuses on four reachability formulas, gives the solution of their satisfiability problems, and analyzes their decidability problems. As an application, an example is provided to show that the termination of the generalized quantum loop program corresponds to the future reachability of Exogenous quantum Markov chains, and therefore can be decided by checking satisfaction of quantum formulas.