Satellite communications (SATCOM) are essential parts of future wireless networks and are moving towards standardization in 6G. Low Earth Orbit (LEO) SATCOM show significant performance gain in global wireless access services, due to their minor path loss and short propagation delay, compared with other SATCOM counterparts. A quality of service (QoS)-aware precoding design is presented in this work to enhance the performance of massive multiple-input multiple-output (MIMO) LEO SATCOM. In particular, we set up a multi-objective optimization problem to balance the system energy efficiency (EE) and the percentage of users achieving their QoS requirements. An efficient algorithm is developed for obtaining the precoding vectors. Numerical results show that the proposed algorithm leads to a larger percentage of users achieving their QoS requirements and brings improvements in EE performance, compared with conventional approaches.