Cloud-based Internet of Things (IoT)-enabled smart healthcare plays a vital role in modern society, yet security and privacy challenges remain unavoidable. The authenticated key agreement (AKA) process, which serves as the foundation of secure communication, is widely recognized as a key solution to these challenges. However, many existing AKA methods in the literature either involve high communication and computational costs or fail to withstand quantum attacks. Post-quantum cryptography (PQC) introduces a new class of cryptographic algorithms designed to resist future quantum computer threats. In this article, we present a quantum-secure, lattice-based three-party AKA scheme for smart IoT healthcare applications, leveraging the computationally complex Ring-Learning With Errors (Ring-LWE) problem. Our approach integrates secure big-data analytics with blockchain technology by utilizing authentication procedures for secure data aggregation before storing it in the blockchain. A comprehensive security evaluation including formal and informal analysis, demonstrates the scheme's strong resilience against both classical and quantum attacks. Additionally, experimental results confirm that the proposed scheme is well-suited for real-time smart healthcare applications.