Quantization is an effective way to compress neural networks. By reducing the bit width of the parameters, the processing efficiency of neural network models at edge devices can be notably improved. Most conventional quantization methods utilize real datasets to optimize quantization parameters and fine-tune. Due to the inevitable privacy and security issues of real samples, the existing real-data-driven methods are no longer applicable. Thus, a natural method is to introduce synthetic samples for zero-shot quantization (ZSQ). However, the conventional synthetic samples fail to retain the detailed texture feature distributions, which severely limits the knowledge transfer and performance of the quantized model. In this paper, a novel ZSQ method, TexQ is proposed to address this issue. We first synthesize a calibration image and extract its calibration center for each class with a texture feature energy distribution calibration method. Then, the calibration centers are used to guide the generator to synthesize samples. Finally, we introduce the mixup knowledge distillation module to diversify synthetic samples for fine-tuning. Extensive experiments on CIFAR10/100 and ImageNet show that TexQ is observed to perform state-of-the-art in ultra-low bit width quantization. For example, when ResNet-18 is quantized to 3-bit, TexQ achieves a 12.18% top-1 accuracy increase on ImageNet compared to state-of-the-art methods. Code at https://github.com/dangsingrue/TexQ.