The Internet of Things (IoT) has transformed healthcare, facilitating remote patient monitoring, enhanced medication adherence, and chronic disease management. However, this interconnected ecosystem faces significant vulnerabilities with the advent of quantum computing, which threatens to break existing encryption standards protecting sensitive patient data in IoT-enabled medical devices. This chapter examines the quantum threat to healthcare IoT security, highlighting the potential impacts of compromised encryption, including privacy breaches, device failures, and manipulated medical records. It introduces post-quantum cryptography (PQC) and quantum-resistant techniques like quantum key distribution (QKD), addressing their application in resource-constrained healthcare IoT devices such as pacemakers, monitoring tools, and telemedicine systems. The chapter further explores the challenges of integrating these solutions and reviews global efforts in mitigating quantum risks, offering insights into suitable PQC primitives for various healthcare use cases.