Vision Transformer (ViT)-based models have shown state-of-the-art performance (e.g., accuracy) in vision-based AI tasks. However, realizing their capability in resource-constrained embedded AI systems is challenging due to their inherent large memory footprints and complex computations, thereby incurring high power/energy consumption. Recently, Spiking Vision Transformer (SViT)-based models have emerged as alternate low-power ViT networks. However, their large memory footprints still hinder their applicability for resource-constrained embedded AI systems. Therefore, there is a need for a methodology to compress SViT models without degrading the accuracy significantly. To address this, we propose QSViT, a novel design methodology to compress the SViT models through a systematic quantization strategy across different network layers. To do this, our QSViT employs several key steps: (1) investigating the impact of different precision levels in different network layers, (2) identifying the appropriate base quantization settings for guiding bit precision reduction, (3) performing a guided quantization strategy based on the base settings to select the appropriate quantization setting, and (4) developing an efficient quantized network based on the selected quantization setting. The experimental results demonstrate that, our QSViT methodology achieves 22.75% memory saving and 21.33% power saving, while also maintaining high accuracy within 2.1% from that of the original non-quantized SViT model on the ImageNet dataset. These results highlight the potential of QSViT methodology to pave the way toward the efficient SViT deployments on resource-constrained embedded AI systems.