End-to-end neural network models achieve improved performance on various automatic speech recognition (ASR) tasks. However, these models perform poorly on edge hardware due to large memory and computation requirements. While quantizing model weights and/or activations to low-precision can be a promising solution, previous research on quantizing ASR models is limited. In particular, the previous approaches use floating-point arithmetic during inference and thus they cannot fully exploit efficient integer processing units. Moreover, they require training and/or validation data during quantization, which may not be available due to security or privacy concerns. To address these limitations, we propose an integer-only, zeroshot quantization scheme for ASR models. In particular, we generate synthetic data whose runtime statistics resemble the real data, and we use it to calibrate models during quantization. We apply our method to quantize QuartzNet, Jasper, and Conformer and show negligible WER degradation as compared to the full-precision baseline models, even without using any data. Moreover, we achieve up to 2.35× speedup on a T4 GPU and 4× compression rate, with a modest WER degradation of <1% with INT8 quantization.