In this letter, we study an unmanned aerial vehicle (UAV)-mounted mobile edge computing network, where the UAV executes computational tasks offloaded from mobile terminal users (TUs) and the motion of each TU follows a Gauss-Markov random model. To ensure the quality-of-service (QoS) of each TU, the UAV with limited energy dynamically plans its trajectory according to the locations of mobile TUs. Towards this end, we formulate the problem as a Markov decision process, wherein the UAV trajectory and UAV-TU association are modeled as the parameters to be optimized. To maximize the system reward and meet the QoS constraint, we develop a QoS-based action selection policy in the proposed algorithm based on double deep Q-network. Simulations show that the proposed algorithm converges more quickly and achieves a higher sum throughput than conventional algorithms.