Recently, fine-tuning large-scale pre-trained GNNs has yielded remarkable attention in adapting pre-trained GNN models for downstream graph learning tasks. One representative fine-tuning method is to exploit adapter (termed AdapterGNN) which aims to 'augment' the pre-trained model by inserting a lightweight module to make the 'augmented' model better adapt to the downstream tasks. However, graph data may contain various types of noise in downstream tasks, such as noisy edges and ambiguous node attributes. Existing AdapterGNNs are often prone to graph noise and exhibit limited generalizability. How to enhance the robustness and generalization ability of GNNs' fine tuning remains an open problem. In this paper, we show that the above problem can be well addressed by integrating uncertainty learning into the GNN adapter. We propose the Uncertainty-aware Adapter (UAdapterGNN) that fortifies pre-trained GNN models against noisy graph data in the fine-tuning process. Specifically, in contrast to regular AdapterGNN, our UAdapterGNN exploits Gaussian probabilistic adapter to augment the pre-trained GNN model. In this way, when the graph contains various noises,our method can automatically absorb the effects of changes in the variances of the Gaussian distribution, thereby significantly enhancing the model's robustness. Also, UAdapterGNN can further improve the generalization ability of the model on the downstream tasks. Extensive experiments on several benchmarks demonstrate the effectiveness, robustness and high generalization ability of the proposed UAdapterGNN method.