The privacy threat, cross-border regulatory differences, and emerging quantum threat are critical challenges to secure cross-border medical data exchange. The present paper suggests a post-quantum cryptography scheme based on lattice-based schemes (LWE, Ring-LWE, Module-LWE) and code-based schemes (McEliece) in order to provide long-term security. The aim is to come up with a robust architecture that ensures confidentiality, integrity and interoperability among global healthcare systems. The implementation combines hybrid-encryption, key-encapsulation schemes, and quantum-resistant, quantum-digital signatures along with secure data transmission protocols. Experimental results demonstrate an improvement in the attack resistance of 53.04 percent and a decrease in the probability of data breach by 85.19 percent over classical cryptographic systems with a reasonable computational overhead (17.94 percent increase). The framework is resistant to algorithmic threats by Shor and Grover. The research finds that future-proof medical data exchange is imperative and can be achieved with post-quantum solutions that proposed scalable and secure interoperability of healthcare across the globe.