Generative Artificial Intelligence (GenAI) offers transformative potential for Government-to-Government (G2G) governance, enhancing the efficiency, transparency, and collaboration between government agencies. This paper explores the applications, benefits, challenges, and ethical considerations of GenAI in G2G contexts, supported by quantitative analysis, real-world case studies, and use cases. By focusing on key areas such as inter-agency data integration, decision support, policy analysis, and crisis management, this paper illustrates how governments can leverage GenAI to streamline operations and create more integrated, responsive public administration systems. The research highlights specific use cases that demonstrate the positive impact of GenAI, including improved processing times, cost savings, and enhanced inter-agency collaboration.