In this paper, we propose an approach for real-time collision-free trajectory planning of multiple robot manipulators in a common workspace. In recent years, robot arms are often introduced to factories in place of human beings, and it has become important how efficiently multiple robot arms can be operated in a small space. The problem of trajectory planning for multiple robot arms is often solved by graph search algorithms, however, it is difficult for the conventional approach to provide flexible trajectory planning to cope with unexpected situations such as robot arm failure. Therefore, we propose a combined method for Q-learning and RRT for the trajectory planning problem. The effectiveness of the proposed method is further verified using numerical experiments. The planned trajectories is able to guarantee a certain degree of optimality when the motion trajectory is generated by combining reinforcement learning than by using only the graph search algorithm. The results indicate that the time required to generate the motion trajectory is reduced by the proposed method.