The intelligent behavior of robots does not emerge solely from control systems, but from the tight coupling between body and brain, a principle known as embodied intelligence. Designing soft robots that leverage this interaction remains a significant challenge, particularly when morphology and control require simultaneous optimization. A significant obstacle in this co-design process is that morphological evolution can disrupt learned control strategies, making it difficult to reuse or adapt existing knowledge. We address this by develop a Graph Neural Network-based approach for the co-design of morphology and controller. Each robot is represented as a graph, with a graph attention network (GAT) encoding node features and a pooled representation passed through a multilayer perceptron (MLP) head to produce actuator commands or value estimates. During evolution, inheritance follows a topology-consistent mapping: shared GAT layers are reused, MLP hidden layers are transferred intact, matched actuator outputs are copied, and unmatched ones are randomly initialized and fine-tuned. This morphology-aware policy class lets the controller adapt when the body mutates. On the benchmark, our GAT-based approach achieves higher final fitness and stronger adaptability to morphological variations compared to traditional MLP-only co-design methods. These results indicate that graph-structured policies provide a more effective interface between evolving morphologies and control for embodied intelligence.