This paper presents a novel, multi-stage modelling approach for integrating Generative AI (GenAI) tools into design-based STEM education, promoting sustainability and 21st-century problem-solving skills. The proposed methodology includes (i) a conceptual model that defines structural aspects of the domain at a high abstraction level; (ii) a contextual model for defining the internal context; (iii) a GenAI-based model for solving the STEM task, which consists of a generic model for integrating GenAI tools into STEM-driven education and a process model, presenting learning/design processes using those tools. A case study involving the design of an autonomous folkrace robot illustrates the implementation of the approach. Based on Likert-scale evaluations, quantitative results demonstrate a significant impact of GenAI tools in enhancing critical thinking, conceptual understanding, creativity, and engineering practices, particularly during the prototyping and testing phases. This paper concludes that the structured integration of GenAI tools supports personalized, inquiry-based, and sustainable STEM education, while also raising new challenges in prompt engineering and ethical use. This approach provides educators with a systematic pathway for leveraging AI to develop STEM-based skills essential for future sustainable development.