Graph neural networks (GNNs) have achieved great success in dealing with graph-structured data that are prevalent in the real world. The core of graph neural networks is the message passing mechanism that aims to generate the embeddings of nodes by aggregating the neighboring node information. However, recent work suggests that GNNs also suffer the trustworthiness issues. Our empirical study shows that the calibration error of the in-distribution (ID) nodes would be exacerbated if a graph is mixed with out-of-distribution (OOD) nodes, and we assume that the noisy information from OOD nodes is the root for the worsened calibration error. Both previous study and our empirical study suggest that adjusting the weights of edges could be a promising way to reduce the adverse impact from the OOD nodes. However, how to precisely select the desired edges and modify the corresponding weights is not trivial, since the distribution of OOD nodes is unknown to us. To tackle this problem, we propose a Graph Edge Re-weighting via Deep Q-learning (GERDQ) framework to calibrate the graph neural networks. Our framework aims to explore the potential influence of the change of the edge weights on target ID nodes by sampling and traversing the edges in the graph, and we formulate this process as a Markov Decision Process (MDP). Many existing GNNs could be seamlessly incorporated into our framework. Experimental results show that when wrapped with our method, the existing GNN models can yield lower calibration error under OOD nodes as well as comparable accuracy compared to the original ones and other strong baselines. The source code is available at:https://github.com/DamoSWL/Calibration-GNN-OOD.