Federated Learning (FL) enables collaborative model training while preserving data privacy, but its classical cryptographic underpinnings are vulnerable to quantum attacks. This vulnerability is particularly critical in sensitive domains like healthcare. This paper introduces PQS-BFL (Post-Quantum Secure Blockchain-based Federated Learning), a framework integrating post-quantum cryptography (PQC) with blockchain verification to secure FL against quantum adversaries. We employ ML-DSA-65 (a FIPS 204 standard candidate, formerly Dilithium) signatures to authenticate model updates and leverage optimized smart contracts for decentralized validation. Extensive evaluations on diverse datasets (MNIST, SVHN, HAR) demonstrate that PQS-BFL achieves efficient cryptographic operations (average PQC sign time: 0.65 ms, verify time: 0.53 ms) with a fixed signature size of 3309 Bytes. Blockchain integration incurs a manageable overhead, with average transaction times around 4.8 s and gas usage per update averaging 1.72 x 10^6 units for PQC configurations. Crucially, the cryptographic overhead relative to transaction time remains minimal (around 0.01-0.02% for PQC with blockchain), confirming that PQC performance is not the bottleneck in blockchain-based FL. The system maintains competitive model accuracy (e.g., over 98.8% for MNIST with PQC) and scales effectively, with round times showing sublinear growth with increasing client numbers. Our open-source implementation and reproducible benchmarks validate the feasibility of deploying long-term, quantum-resistant security in practical FL systems.