Owing to the expansion of non-face-to-face activities, security issues in video conferencing systems are becoming more critical. In this paper, we focus on the end-to-end encryption (E2EE) function among security services of video conferencing systems. First, the E2EE-related protocols of Zoom and Secure Frame (SFrame), which are representative video conferencing systems, are thoroughly investigated, and the two systems are compared and analyzed overall. Next, the E2EE protocol in the Government Public Key Infrastructure (GPKI)-based video conferencing system, in which the user authentication mechanism is fundamentally different from those used in private sector systems, such as Zoom or SFrame, will be considered. In particular, among E2EE-related protocols, we propose a detailed mechanism in which the post-quantum cryptography (PQC) key encapsulation mechanism (KEM) is applied to the user key exchange process. As the session key is not disclosed to the central server even in futuristic quantum computers, the mechanism to which the PQC KEM method proposed in this study is applied satisfies the E2EE security requirements and can be applied to domestic GPKI-based systems. It is expected to contribute in strengthening the safety of the next-generation video conferencing system.