Large Language Models (LLMs) fine-tuning technologies have achieved remarkable results. However, traditional LLM fine-tuning approaches face significant challenges: they require large Floating Point (FP) computation, raising privacy concerns when handling sensitive data, and are impractical for resource-constrained edge devices. While Parameter-Efficient Fine-Tuning (PEFT) techniques reduce trainable parameters, their reliance on floating-point arithmetic creates fundamental incompatibilities with edge hardware. In this work, we introduce a novel framework for on-device LLM fine-tuning that eliminates the need for floating-point operations in both inference and training, named GSQ-Tuning. At its core is the Group-Shared Exponents Integer format, which efficiently represents model parameters in integer format using shared exponents among parameter groups. When combined with LoRA-like adapters, this enables fully integer-based fine-tuning that is both memory and compute efficient. We demonstrate that our approach achieves accuracy comparable to BF16-based fine-tuning while significantly reducing 1.85x memory usage. Moreover, compared to FP8, our method can reduce 5x power consumption and 11x chip area with same performance, making large-scale model adaptation feasible on edge devices.