Recent advancements in computational chemistry have increasingly focused on synthesizing molecules based on textual instructions. Integrating graph generation with these instructions is complex, leading most current methods to use molecular sequences with pre-trained large language models. In response to this challenge, we propose a novel framework, named \(\textbf{UTGDiff (Unified Text-Graph Diffusion Model)}\), which utilizes language models for discrete graph diffusion to generate molecular graphs from instructions. UTGDiff features a unified text-graph transformer as the denoising network, derived from pre-trained language models and minimally modified to process graph data through attention bias. Our experimental results demonstrate that UTGDiff consistently outperforms sequence-based baselines in tasks involving instruction-based molecule generation and editing, achieving superior performance with fewer parameters given an equivalent level of pretraining corpus. Our code is availble at https://github.com/ran1812/UTGDiff.