Fifth Generation (5G) cellular networks have been adopted worldwide since the rollout began around 2019. It brought with it many innovations and new services, such as Enhanced Mobile Broadband (eMBB), Ultra Reliable and Low-Latency Communications (URLLC), and Massive Internet of Things (mIoT). Furthermore, 5G introduced a more scalable approach to network operations using fully software-based Virtualized Network Functions (VNF) in Core Networks (CN) rather than the prior hardware-based approach. However, while this shift towards a fully software-based system design provides numerous significant benefits, such as increased interoperability, scalability, and cost-effectiveness, it also brings with it an increased cybersecurity risk. Security is crucial to maintaining trust between vendors, operators, and consumers. Cyberattacks are rapidly increasing in number and sophistication, and we are seeing a shift towards zero-trust approaches. This means that even communications between VNFs inside a 5G core must be scrutinized and hardened against attacks, especially with the advent of quantum computers. The National Institute of Standards and Technology (NIST), over the past 10 years, has led efforts to standardize post-quantum cryptography (PQC) to protect against quantum attacks. This paper covers a custom implementation of the open-source free5GC CN, to expand its HTTPS capabilities for VNFs by introducing PQC Key Encapsulation Methods (KEM) for Transport Layer Security (TLS) v1.3. This paper provides the details of this integration with a focus on the latency of different PQC KEMs in initial handshakes between VNFs, on packet size, and the implications in a 5G environment. This work also conducts a security comparison between the PQC-equipped free5GC and other open-source 5G CNs. The presented results indicate a negligible increase in UE connection setup duration and a small increase in connection setup data requirements, strongly indicating that PQC KEM's benefits far outweigh any downsides when integrated into 5G and 6G core services. To the best of our knowledge, this is the first work incorporating PQC into an open-source 5G core. Furthermore, the results from this effort demonstrate that employing PQC ciphers for securing VNF communications results in only a negligible impact on latency and bandwidth usage, thus demonstrating significant benefits to 5G cybersecurity. [ABSTRACT FROM AUTHOR]