In this paper, we investigate the task of quality of service (QoS) routing in software-defined networks (SDN). We consider delay, bandwidth, loss, and cost as QoS parameters. We propose a new deep reinforcement learning solution for greedy online QoS routing in SDN and call it Deep Q-Routing (DQR). DQR utilises a dueling deep Q-network with prioritised experience replay to compute a path for any source-destination pair request in the presence of multiple QoS metrics. In contrast to existing DRL-based routing methods, the proposed DQR method regards the task of routing as a discrete control problem and uses a reward function comprising weighted QoS parameters. Our simulation results show that DQR substantially improves end-to-end throughput compared to other existing learning-based methods.