Relational learning has gained significant attention, led by the expressiveness of Graph Neural Networks (GNNs) on graph data. While the inherent biases in common graph data are involved in GNN training, it poses a serious challenge to constraining the GNN output perturbations induced by input biases, thereby safeguarding fairness during training. The Lipschitz constant, a technique from robust statistics, can limit the maximum changes in the output concerning the input, taking into account associated irrelevant biased factors. It is an efficient and provable method to examine the output stability of machine learning models without incurring additional computational costs. Recently, its use in controlling the stability of Euclidean neural networks, the calculation of the precise Lipschitz constant remains elusive for non-Euclidean neural networks like GNNs, especially within fairness contexts. However, no existing research has investigated Lipschitz constants to shed light on stabilizing the GNN outputs, especially when working on graph data with implicit biases. To narrow this gap, we begin with the general GNNs operating on an attributed graph, and formulate a Lipschitz constant to limit the changes in the output regarding biases associated with the input. Additionally, we theoretically analyze how the Lipschitz constant of a GNN model could constrain the output perturbations induced by biases learned from data for fairness training. We experimentally validate the Lipschitz constant's effectiveness in limiting biases of the model output. Finally, from a training dynamics perspective, we demonstrate why the theoretical Lipschitz constant can effectively guide the GNN training to better trade-off between accuracy and fairness.