Quantization reduces the deployment cost of large language models (LLMs). However, post-training quantization (PTQ) cannot fine-tune model parameters and often degrades accuracy in low-bit settings, while quantization-aware training (QAT) requires backpropagation and incurs high memory overhead. We propose ZeroQAT, a zeroth-order QAT framework supporting both weight and activation quantization. By using forward-only gradient estimation, ZeroQAT removes backpropagation and substantially reduces memory and computation while retaining end-to-end optimization benefits. We further introduce a lightweight variant of ZeroQAT for quantized fine-tuning, which freezes and pre-quantizes most parameters to further cut memory usage. Experiments show that ZeroQAT consistently outperforms representative PTQ and QAT baselines while requiring significantly less memory. For example, ZeroQAT enables fine-tuning of a 13B model at extremely low bit-widths (e.g., 2-4 bits) on a single 8GB GPU, and even allows fine-tuning a 6.7B model on a OnePlus 12 smartphone, demonstrating its practicality for end-to-end QAT on resource-limited edge devices.