Searching and detecting communities in real-world graphs underpins a wide range of applications. Despite the success achieved, current learning-based solutions regard community search, i.e., locating the best community for a given query, and community detection, i.e., partitioning the whole graph, as separate problems, necessitating task- and dataset-specific retraining. Such a strategy limits the applicability and generalization ability of the existing models. Additionally, these methods rely heavily on information from the target dataset, leading to suboptimal performance when supervision is limited or unavailable. To mitigate this limitation, we propose UniCom, a unified framework to solve both community search and detection tasks through knowledge transfer across multiple domains, thus alleviating the limitations of single-dataset learning. UniCom centers on a Domain-aware Specialization (DAS) procedure that adapts on the fly to unseen graphs or tasks, eliminating costly retraining while maintaining framework compactness with a lightweight prompt-based paradigm. This is empowered by a Universal Graph Learning (UGL) backbone, which distills transferable semantic and topological knowledge from multiple source domains via comprehensive pre-training. Both DAS and UGL are informed by local neighborhood signals and cohesive subgraph structures, providing consistent guidance throughout the framework. Extensive experiments on both tasks across 16 benchmark datasets and 22 baselines have been conducted to ensure a comprehensive and fair evaluation. UniCom consistently outperforms all state-of-the-art baselines across all tasks under settings with scarce or no supervision, while maintaining runtime efficiency.