Large Language Models (LLMs) have demonstrated remarkable performance in code intelligence tasks such as code generation, summarization, and translation. However, their reliance on linearized token sequences limits their ability to understand the structural semantics of programs. While prior studies have explored graphaugmented prompting and structure-aware pretraining, they either suffer from prompt length constraints or require task-specific architectural changes that are incompatible with large-scale instructionfollowing LLMs. To address these limitations, this paper proposes CGBridge, a novel plug-and-play method that enhances LLMs with Code Graph information through an external, trainable Bridge module. CGBridge first pre-trains a code graph encoder via selfsupervised learning on a large-scale dataset of 270K code graphs to learn structural code semantics. It then trains an external module to bridge the modality gap among code, graph, and text by aligning their semantics through cross-modal attention mechanisms. Finally, the bridge module generates structure-informed prompts, which are injected into a frozen LLM, and is fine-tuned for downstream code intelligence tasks. Experiments show that CGBridge achieves notable improvements over both the original model and the graphaugmented prompting method. Specifically, it yields a 16.19% and 9.12% relative gain in LLM-as-a-Judge on code summarization, and a 9.84% and 38.87% relative gain in Execution Accuracy on code translation. Moreover, CGBridge achieves over 4x faster inference than LoRA-tuned models, demonstrating both effectiveness and efficiency in structure-aware code understanding.