Large Language Model (LLM) routing has demonstrated strong capability in balancing response quality with computational cost. As users exhibit diverse preferences, personalization has attracted increasing attention in LLM routing, since even identical queries may require different models to generate responses tailored to individual needs. However, existing approaches are not fully personalized and often fail to capture the complex interactions between specific users and LLMs. Moreover, user preference data is typically scarce, noisy, and inconsistent in format, which limits the effectiveness of methods that rely solely on user-specific data. To address these challenges, we propose GMTRouter, which represents multi-turn user-LLM interactions as a heterogeneous graph with four node types: user, LLM, query, and response, thereby preserving the rich relational structure of the interaction. Through a tailored message-passing mechanism, GMTRouter learns to capture user preferences from few-shot data within a lightweight inductive graph learning framework, enabling effective personalization. Extensive experiments demonstrate that GMTRouter consistently outperforms strong baselines, achieving 0.9 to 21.6 percent higher accuracy and 0.006 to 0.309 higher AUC across multiple datasets. More importantly, we demonstrate that GMTRouter can adapt to new users and evolving preferences using only few-shot data, without extensive fine-tuning. The code for GMTRouter is publicly available at https://github.com/ulab-uiuc/GMTRouter.