With the construction of large-scale Low Earth Orbit (LEO) satellite constellations, providing worldwide internet service has become a reality. The wide-coverage and openness featured in LEO satellite networks introduce the security problem. Thus, determined efforts have been made to integrate quantum communications with LEO satellite networks to secure satellite communications, yet with only one hop. However, securing LEO satellite networks calls for multi-hop quantum link establishment based on quantum entanglements, which remains an open problem. For the first time, we present a highly efficient routing scheme named Q-Star for LEO satellite quantum networks, with the objective of building long-distance concurrent end-to-end quantum entanglements. However, building multiple long-distance concurrent quantum links is challenging under the high dynamics of LEO satellite networks, which involves real-time quantum link scheduling and qubit resource allocation. To address this challenge, we propose several novel algorithms such as Q-Floyd, redundant path reservation, geocentric-based link status collection and max-fair quantum link allocation which account for the unique features such as probability and exclusivity of link establishment in quantum communications. Finally, we have developed a simulation prototype for LEO satellite quantum networks driven by real LEO satellite constellations and traces. Experiment results verify that the proposed routing algorithm can considerably improve the throughput compared to other strategies. This work is expected to provide a guideline for promoting the integration of LEO satellite networks with quantum communications.