As a key paradigm of future 6G networks, Space-Air-Ground Integrated Networks (SAGIN) has been envisioned to provide numerous intelligent applications that necessitate the cooperation of a multitude of terrestrial devices for machine learning (ML) model training. Utilizing the satellite as the central server, federated learning (FL) offers a promising strategy for distributed training with enhanced data security and privacy. However, employing FL in SAGIN faces challenges in managing vast datasets, training complicated ML models, and ensuring security in long-distance transmission of ML models. In this article, we propose a quantum-empowered FL framework integrating variational quantum algorithms (VQA) and quantum relays in SAGIN. Our approach employs VQA-based ML for local training of FL, addressing the complexity emerging from vast datasets and ML models. Furthermore, supported by unmanned aerial vehicles (UAVs) and high-altitude platform stations (HAPS), the proposed quantum relay scheme via quantum teleportation guarantees security in long-distance model transmission. We present a case study to validate the proposed VQA-based local training and quantum relaying model transmission. The numerical results demonstrate the feasibility and efficiency of the proposed VQA-based FL framework and the quantum relay-based model transmission. Our approach highlights the potential of integrating quantum techniques with FL in SAGIN, which can enable secure, efficient, and advanced edge intelligence applications.