In the upcoming 6G era, existing terrestrial networks have evolved toward space-air-ground integrated networks (SAGIN), providing ultra-high data rates, seamless network coverage, and ubiquitous intelligence for communications of applications and services. However, conventional communications in SAGIN still face data confidentiality issues. Fortunately, the concept of Quantum Key Distribution (QKD) over SAGIN is able to provide information-theoretic security for secure communications in SAGIN with quantum cryptography. Therefore, in this paper, we propose the quantum-secured SAGIN which is feasible to achieve proven secure communications using quantum mechanics to protect data channels between space, air, and ground nodes. Moreover, we propose a universal QKD service provisioning framework to minimize the cost of QKD services under the uncertainty and dynamics of communications in quantumsecured SAGIN. In this framework, fiber-based QKD services are deployed in passive optical networks with the advantages of low loss and high stability. Moreover, the widely covered and flexible satellite-and UAV-based QKD services are provisioned as a supplement during the real-time data transmission phase. Finally, to examine the effectiveness of the proposed concept and framework, a case study of quantum-secured SAGIN in the Metaverse is conducted where uncertain and dynamic factors of the secure communications in Metaverse applications are effectively resolved in the proposed framework.