In this paper, we demonstrate that a Quantum Vision Transformer (QViT) with quantum self-attention (QSA) mechanism outperforms the state-of-the-art (SOTA) biomedical image classifier while using 99.99% fewer parameters. We replace the linear self-attention (SA) layer with a parameterized quantum neural network (QNN) to create the QSA mechanism, reducing the parameter size from O(n²) to O(n). On the RetinaMNIST dataset, QViT outperforms 13 out of 14 SOTA methods, including CNNs and ViT, achieving an accuracy of 56.5%, which is 0.88% lower than the state-of-the-art model, MedMamba, which uses 14.5M parameters, yet uses 99.99% fewer parameters (1K vs 14.5M) and 89% fewer GFLOPs. Furthermore, we apply knowledge distillation (KD) from classical vision transformers to quantum vision transformers for the first time to biomedical image classification, showing that QViT improves QSA parameter efficiency while maintaining comparable performance to classical ViT on eight datasets with various modalities. High-qubit architectures benefit more from KD pre-training, suggesting a scaling relationship between QSA parameters and KD effects. These results establish QSA as a practical architecture choice for parameter-efficient biomedical image analysis.