While Generative AI (GenAI) systems draw users away from (Q&A) forums, they also depend on the very data those forums produce to improve their performance. Addressing this paradox, we propose a framework of sequential interaction, in which a GenAI system proposes questions to a forum that can publish some of them. Our framework captures several intricacies of such a collaboration, including non-monetary exchanges, asymmetric information, and incentive misalignment. We bring the framework to life through comprehensive, data-driven simulations using real Stack Exchange data and commonly used LLMs. We demonstrate the incentive misalignment empirically, yet show that players can achieve roughly half of the utility in an ideal full-information scenario. Our results highlight the potential for sustainable collaboration that preserves effective knowledge sharing between AI systems and human knowledge platforms.