The increasing frequency of extreme weather events, particularly droughts, poses significant challenges to the resilience and sustainability of rural microgrids. These systems, which integrate renewable energy and water resources to support agricultural operations such as dairy farming and greenhouse cultivation, are especially vulnerable due to resource scarcity and uncertainty. This paper proposes a novel optimization framework that combines quantum computing-based distributionally robust optimization (DRO) with classical optimization techniques to address these challenges. The hybrid quantum-classical DRO approach is designed to manage the complex interdependencies between energy and water resources, ensuring robust performance under uncertainties. The proposed model optimizes resource allocation and balances operational costs with resilience metrics such as renewable energy utilization and water use efficiency. By considering the uncertainties in renewable energy generation, water availability, and drought-induced scarcity, the framework provides a comprehensive solution for rural microgrid operation. The optimization process is demonstrated through case studies of a rural microgrid supporting dairy farms and greenhouses, where it effectively reduces operational costs, improves system efficiency, and enhances resilience.