This study presents a complete quantum-aware Internet of Medical Things (IoMT) framework for secure medical imaging that integrates lightweight edge diagnostics, quantum key distribution, and hybrid image encryption. At the edge layer, MicroRadNet, an ultra-compact convolutional neural network (CNN) with exactly 3,140 trainable parameters, is deployed on a Raspberry Pi Zero 2 W, achieving 98.24% accuracy on chest X-ray classification with INT8 quantization. This reduces the model size to 3.14 KB and yields inference latency that meets real-time constraints. At the communication layer, a gateway-mediated BB84 protocol executed on Amazon Braket generates session keys via Golay [24, 12, 8] reconciliation (correcting up to t = 3 errors per codeword) and privacy amplification, producing 256-bit keys with quantum bit error rate (QBER) < 0.02. At the image-protection layer, a hybrid cipher combines 5D hyperchaotic permutation (seeded from BB84 key material), fixed-rule DNA encoding, and quantum-keyed diffusion. Security evaluation demonstrates near-ideal entropy, strong NPCR and UACI values, uniform ciphertext histogram distribution, low directional adjacent-pixel correlations, low average absolute correlation, large key space, high plaintext sensitivity, high key sensitivity, and stable chaotic-sequence randomness. These results provide empirical statistical validation of resistance to common statistical, differential, brute-force, and key-sensitivity attacks. End-to-end latency remains below the 1.5 s system constraint, validating practical edge deployment. The framework replaces classical public-key dependency with simulation-validated quantum-aware foundations while maintaining clinical accuracy and real-time performance, establishing a path toward deployable, standards-aligned quantum-aware healthcare security.