Electroencephalography (EEG) is a non-invasive and cost-effective tool for capturing brain signals. However, existing approaches based on deep learning models for classifying EEG signals are primarily trained on large EEG datasets and require extensive computational resources. This poses significant challenges for real-time processing, particularly in resource-constrained environments such as wearable devices and edge computing. Our work addresses these challenges by exploring quantization techniques to optimize a vision transformer model for real-time EEG classification. We apply post-training quantization to the proposed Vision Transformer (ViT) model, termed the Brain Signal Vision Transformer (BSVT), to reduce computational overhead while maintaining high performance. The BSVT model, trained on the ex3 HPC system, efficiently handles large datasets. We comprehensively analyze the trade-offs between model accuracy and computational efficiency. Specifically, we demonstrate the use of dynamic quantization on a ViT-based model for EEG classification, achieving a 2.49x reduction in model size and a 1.46x improvement in inference speed. These findings underscore the potential of quantization techniques and distributed computing to revolutionize the adoption of AI in healthcare.