The Service-Based Architecture (SBA) of the fifth generation (5G) of cellular networks introduced great advancements in flexibility, efficiency, and performance of modern networks, thanks to the distribution and softwarization of core network entities called Network Functions (NFs). Due to the distributed nature of 5G systems, these NFs are usually located in different sites to provide better Quality of Service (QoS) or specific services, which poses a great challenge regarding security. However, even though the 3GPP standard identifies security requirements by leveraging Transport Layer Security (TLS) authentication and OAuth2.0 authorization, current experimental and commercial deployments often disregard these aspects for the sake of simplicity and performance. In fact, even though data and communication protection is crucial, the implementation of security protocols and mechanisms complicates the deployments and may reduce the efficiency of 5G systems. This paper presents a novel solution that implements TLS for authentication and encryption leveraging entanglement-based Quantum Key Distribution (QKD), ensuring that all the communications between NFs in the 5G core network are secured by QKD keys, and are therefore quantum-safe. Additionally, this paper presents a QKD key repository using the Network Repository Function (NRF), which stores and relays TLS session keys to authenticated consumer NFs to access producer NFs in a fast but quantum-safe process. The solution was validated in a real hardware environment, and obtained results demonstrate that the proposed solution enhances the security of the SBA communications in an efficient way, reducing an 85% the service request time compared to the traditional TLS-based procedure, as well as 29.96% fewer bytes transmitted throughout the process.