Post-Training Quantization enables efficient Vision Transformer (ViTs) deployment with a small calibration data, and its prevalent use of uniform quantization harnesses AI accelerator matrix cores for high-speed inference. However, the application of uniform quantization is fundamentally challenged by the extreme non-uniformity of activation distributions.Specifically, the power-law nature of post-Softmax attention scores and the significant inter-channel variance in post-GELU activations create a dilemma for conventional quantization, as it struggles to preserve critical high-magnitude values without sacrificing overall precision. To resolve this core conflict, we introduce UQ-ViT (Uniform Quantization for Vision Transformers), a novel uniform quantization framework designed to reconcile high precision with hardware efficiency. Central to UQ-ViT are two operators: Dynamic Elimination of Maximum (DeMax) and Normalization Quantization (NormQuant). DeMax is a quantization operator for post-Softmax attention scores that utilizes uniform quantization. It dynamically eliminates and preserves dominant values, effectively mitigating quantization loss from the extreme values in the power-law distribution. NormQuant utilizes a per-channel quantization strategy during quantization and reverts to a per-tensor format for dequantization, achieving both high accuracy and computational efficiency. Crucially, it is applicable to any linear layer, enabling effective quantization of post-GELU activations in ViTs. Through extensive experiments on various ViTs and vision tasks, including image classification, object detection, and instance segmentation, we demonstrate that our proposed approach outperforms existing methods, achieving superior accuracy while ensuring hardware friendliness.