Cube satellites (CubeSats) have grown into the primary non-terrestrial network capable of providing global access services in satellite-air-ground integrated networks (SAGIN). Nonetheless, the provision of genuinely global access services solely via CubeSats is challenging due to the frequent handovers and the existence of polar regions where service availability is compromised in SAGIN. To tackle these issues, the design of innovative quantum multi-agent reinforcement learning (QMARL)-based algorithm is tailored for the cooperative scheduling of multi-CubeSat/high-altitude long-endurance unmanned aerial vehicle (HALE-UAV) systems. This algorithm aims to achieve high quality of services, energy efficiency, and high capacity. Furthermore, logarithmic scale reduction in action dimensions can be realized, due to the modification in quantum measurement in QMARL. This is essential when the number of CubeSats and HALE-UAVs increases. Based on a realistic CubeSat/HALE-UAV experimental environment using real-world data, the excellence of our proposed QMARL-based scheduler is demonstrated.