Graph neural networks (GNNs) are conventionally trained on a per-domain, per-task basis. It creates a significant barrier in transferring the acquired knowledge to different, heterogeneous data setups. This paper introduces GraphBridge, a novel framework to enable knowledge transfer across disparate tasks and domains in GNNs, circumventing the need for modifications to task configurations or graph structures. Specifically, GraphBridge allows for the augmentation of any pre-trained GNN with prediction heads and a bridging network that connects the input to the output layer. This architecture not only preserves the intrinsic knowledge of the original model but also supports outputs of arbitrary dimensions. To mitigate the negative transfer problem, GraphBridge merges the source model with a concurrently trained model, thereby reducing the source bias when applied to the target domain. Our method is thoroughly evaluated across diverse transfer learning scenarios, including Graph2Graph, Node2Node, Graph2Node, and graph2point-cloud. Empirical validation, conducted over 16 datasets representative of these scenarios, confirms the framework's capacity for task- and domain-agnostic transfer learning within graph-like data, marking a significant advancement in the field of GNNs. Code is available at https://github.com/jujulili888/GraphBridge.