Quantized Neural Networks (QNNs) have emerged as a promising solution for reducing model size and computational costs, making them well-suited for deployment in edge and resource-constrained environments. While quantization is known to disrupt gradient propagation and enhance robustness against pixel-level adversarial attacks, its effectiveness against patch-based adversarial attacks remains largely unexplored. In this work, we demonstrate that adversarial patches remain highly transferable across quantized models, achieving over 70\% attack success rates (ASR) even at extreme bit-width reductions (e.g., 2-bit). This challenges the common assumption that quantization inherently mitigates adversarial threats. To address this, we propose Quantization-Aware Defense Training with Randomization (QADT-R), a novel defense strategy that integrates Adaptive Quantization-Aware Patch Generation (A-QAPA), Dynamic Bit-Width Training (DBWT), and Gradient-Inconsistent Regularization (GIR) to enhance resilience against highly transferable patch-based attacks. A-QAPA generates adversarial patches within quantized models, ensuring robustness across different bit-widths. DBWT introduces bit-width cycling during training to prevent overfitting to a specific quantization setting, while GIR injects controlled gradient perturbations to disrupt adversarial optimization. Extensive evaluations on CIFAR-10 and ImageNet show that QADT-R reduces ASR by up to 25\% compared to prior defenses such as PBAT and DWQ. Our findings further reveal that PBAT-trained models, while effective against seen patch configurations, fail to generalize to unseen patches due to quantization shift. Additionally, our empirical analysis of gradient alignment, spatial sensitivity, and patch visibility provides insights into the mechanisms that contribute to the high transferability of patch-based attacks in QNNs.