The rise of quantum computing presents substantial challenges to the security of Peer-to-Peer (P2P) energy trading systems. This paper introduces an innovative mutual authenticated session key agreement protocol aimed at protecting edge-IoT P2P energy trading communications in the quantum era. Each peer is an IoT-enabled smart metering device that is registered by an edge registration authority and acts as both producer and consumer. The proposed protocol incorporates CRYSTALS-Kyber's quantum-resistant cryptographic suite to guarantee the authenticity and confidentiality of negotiated session keys in the presence of quantum threats. Furthermore, it leverages the Hedera Hashgraph public blockchain, which is recognized for its quantum-resilient design, to ensure transparency and support efficient revocation mechanisms. Extensive comparative analyses have been conducted to demonstrate the protocol's superiority over existing alternatives, including both traditional schemes based on classical cryptography and emerging early-stage post-quantum approaches, in terms of performance and security features. By combining robust security and high efficiency, the proposed protocol fosters the development of resilient decentralized energy markets, paving the way for secure and sustainable energy distribution in the future.

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