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Accurate prediction of demand for emergency, imaging, pharmacy, laboratory, and inpatient services is critical for hospital planning. However, forecasting models are typically built separately by department or institution, which limits their ability to learn from shared demand patterns. Hospitals generate rich operational demand streams, but patient-level data cannot usually be pooled across organizations. This creates a need for collaborative forecasting methods that preserve institutional control over sensitive operational records. This article proposes a federated multi-task learning framework for predicting service demand across multiple hospital units. The framework trains a shared predictive model across hospitals while each institution contributes only protected model updates. The framework includes local data adapters, a shared temporal learning backbone, task-specific forecasting heads, a federated aggregation layer, differential privacy mechanisms, and site-specific personalization modules. Together, these components support collaborative forecasting without transferring patient-level operational data. The framework could improve demand prediction by learning common temporal patterns across hospitals and service lines. It would also support local adaptation, reduce duplicated model development, and preserve data confidentiality. A privacy-preserving, collaborative approach to hospital demand forecasting could become a core infrastructure for multi-site operational coordination. Federated multi-task learning offers a practical conceptual foundation for such a system.