The Internet of Medical Things (IoMT) has significantly enhanced the healthcare system by enabling advanced patient monitoring, data analytics, and remote interactions. Given that IoMT devices generate vast amounts of sensitive data, robust privacy mechanisms are essential. This privacy requirement is critical for IoMT as, generally, these devices are very resource-constrained with limited storage, computation, and communication capabilities. Blockchain technology, with its decentralisation, transparency, and immutability, offers a promising solution for improving IoMT data security and privacy. However, the recent emergence of quantum computing necessitates developing measures to maintain the security and integrity of these data against emerging quantum threats. This work addresses the current gap of a comprehensive review and analysis of the research efforts to secure IoMT data using blockchain in the quantum era. We discuss the importance of blockchain for IoMT privacy and analyse the impact of quantum computing on blockchain to justify the need for these works. We also provide a comprehensive review of the existing literature on quantum-resistant techniques for effective blockchain solutions in IoMT applications. From our detailed review, we present challenges and future opportunities for blockchain technology in this domain.