This paper addresses the critical challenge of energy efficiency in distributed Internet of Things (IoT) networks through the application of federated learning-based energy management techniques tailored for green computing. With the exponential growth of connected devices, traditional centralized processing poses significant privacy, communication and energy consumption issues. Federated learning offers a decentralized paradigm that preserves user privacy while enabling collective model training across heterogeneous IoT nodes. This work proposes novel energy-aware federated learning algorithms that optimize communication and computation costs by leveraging techniques such as adaptive model updates, quantization, and device participation scheduling. The proposed framework integrates trust mechanisms to ensure secure and reliable cooperation among devices, thereby enhancing sustainability and network longevity. Experimental evaluations demonstrate significant reductions in energy consumption without compromising learning accuracy, highlighting the potential for real-world implementation in diverse IoT environments. The findings underscore the importance of leveraging collaborative intelligence for sustainable, green computing infrastructures, paving the way for future research in scalable, energy-efficient federated learning applications within IoT networks.