Quantum routing plays a key role in the development of the next-generation network system. In particular, an entangled routing path can be constructed with the help of quantum entanglement and swapping among particles (e.g., photons) associated with nodes in the network. From another side of computing, machine learning has achieved numerous breakthrough successes in various application domains, including networking. Despite its advantages and capabilities, machine learning is not as much utilized in quantum networking as in other areas. To bridge this gap, in this article, we propose a novel quantum routing model for quantum networks that employs machine learning architectures to construct the routing path for the maximum number of demands (source–destination pairs) within a time window. Specifically, we present a deep reinforcement routing scheme that is called Deep Quantum Routing Agent (DQRA). In short, DQRA utilizes an empirically designed deep neural network that observes the current network states to accommodate the network’s demands, which are then connected by a qubit-preserved shortest path algorithm. The training process of DQRA is guided by a reward function that aims toward maximizing the number of accommodated requests in each routing window. Our experiment study shows that, on average, DQRA is able to maintain a rate of successfully routed requests at above 80% in a qubit-limited grid network and approximately 60% in extreme conditions, i.e., each node can be repeater exactly once in a window. Furthermore, we show that the model complexity and the computational time of DQRA are polynomial in terms of the sizes of the quantum networks.