Federated Learning (FL) enables collaborative training of medical AI models across hospitals without centralizing patient data. However, the exchange of model updates exposes critical vulnerabilities: gradient inversion attacks can reconstruct patient information, Byzantine clients can poison the global model, and the \emph{Harvest Now, Decrypt Later} (HNDL) threat renders today's encrypted traffic vulnerable to future quantum adversaries.We introduce \textbf{ZKFL-PQ} (\emph{Zero-Knowledge Federated Learning, Post-Quantum}), a three-tiered cryptographic protocol that hybridizes (i) ML-KEM (FIPS~203) for quantum-resistant key encapsulation, (ii) lattice-based Zero-Knowledge Proofs for verifiable \emph{norm-constrained} gradient integrity, and (iii) BFV homomorphic encryption for privacy-preserving aggregation. We formalize the security model and prove correctness and zero-knowledge properties under the Module-LWE, Ring-LWE, and SIS assumptions \emph{in the classical random oracle model}. We evaluate ZKFL-PQ on synthetic medical imaging data across 5 federated clients over 10 training rounds. Our protocol achieves \textbf{100\% rejection of norm-violating updates} while maintaining model accuracy at 100\%, compared to a catastrophic drop to 23\% under standard FL. The computational overhead (factor $\sim$20$\times$) is analyzed and shown to be compatible with clinical research workflows operating on daily or weekly training cycles. We emphasize that the current defense guarantees rejection of large-norm malicious updates; robustness against subtle low-norm or directional poisoning remains future work.