Generative artificial intelligence (AI) in edge networks has excelled in delivering human-level creative services close to the end users. However, providing customized intelligence services to a wide range of end clients remains challenging due to the diverse demands of edge applications. In this paper, we present FlexGen, an efficient generative AI framework with flexible diffusion models, to tailor the intelligence service for different client-side requests under diverse quality and efficiency constraints. To this end, we first design and train a flexible diffusion model to support quality-and-cost adjustable image synthesis. After that, we focus on the server-side energy minimization problem subject to the quality level of generative service and the client-side latency constraint. We further theoretically characterize the relationship between the width of the diffusion model and the expected quality of the synthetic image. Following that, the decomposition solution is applied to optimize the generative service, where the image synthesis strategy and resource allocation policy are personalized for different client-side requests. Experiments indicate that, compared to existing image generation schemes, our framework can save up to two times energy consumption without sacrificing the quality of the service.