This article investigates the cooperative output regulation problem for discrete-time heterogeneous multiagent Markov jump systems. Two cases are studied: 1) output regulation quadratic control in the case where the exosystem is accessible to all agents and 2) cooperative output regulation quadratic control in the case where only a part of agents can directly communicate with the exosystem. The hidden Markov models are employed to describe the asynchronous modes of the agents and their corresponding controllers. Via the jumping regulator equation, asynchronous control laws are constructed and the algorithms to obtain control parameters are presented in terms of linear matrix inequalities. For the first case, the optimal synchronous/mode-dependent control law, which is a special case of the asynchronous control protocol, is also given via the stochastic dynamic programming approach. Finally, an example is given to illustrate the effectiveness of the proposed approaches.