Diabetes is a chronic disease with a significant global health burden, requiring multi-stakeholder collaboration for optimal management. Large language models (LLMs) have shown promise in various healthcare scenarios, but their effectiveness across diverse diabetes tasks remains unproven. Our study introduced a framework to train and validate diabetes-specific LLMs. We first developed a comprehensive data processing pipeline that includes data collection, filtering, augmentation and refinement. This created a high-quality, diabetes-specific dataset and evaluation benchmarks from scratch. Fine-tuned on the collected training dataset, our diabetes-specific LLM family demonstrated state-of-the-art proficiency in processing various diabetes tasks compared to other LLMs. Furthermore, clinical studies revealed the potential applications of our models in diabetes care, including providing personalized healthcare, assisting medical education, and streamlining clinical tasks. Generally, our introduced framework helps develop diabetes-specific LLMs and highlights their potential to enhance clinical practice and provide personalized, data-driven support for diabetes management across different end users. Our codes, benchmarks and models are available at https://github.com/waltonfuture/Diabetica.