Recently, generative AI technologies have emerged as a significant advancement in artificial intelligence field, renowned for their language and image generation capabilities. Meantime, space-air-ground integrated network (SAGIN) is an integral part of future B5G/6G for achieving ubiquitous connectivity. Inspired by this, this article explores an integration of generative AI in SAGIN, focusing on potential applications and case study. We first provide a comprehensive review of SAGIN and generative AI models, highlighting their capabilities and opportunities of their integration. Benefiting from generative AI's ability to generate useful data and facilitate advanced decision-making processes, it can be applied to various scenarios of SAGIN. Accordingly, we present a concise survey on their integration, including channel modeling and channel state information (CSI) estimation, joint air-space-ground resource allocation, intelligent network deployment, semantic communications, image extraction and processing, security and privacy enhancement. Next, we propose a framework that utilizes a Generative Diffusion Model (GDM) to construct channel information map to enhance quality of service for SAGIN. Simulation results demonstrate the effectiveness of the proposed framework. Finally, we discuss potential research directions for generative AI-enabled SAGIN.