Conventional graph neural networks (GNNs) are often confronted with fairness issues that may stem from their input, including node attributes and neighbors surrounding a node. While several recent approaches have been proposed to eliminate the bias rooted in sensitive attributes, they ignore the other key input of GNNs, namely the neighbors of a node, which can introduce bias since GNNs hinge on neighborhood structures to generate node representations. In particular, the varying neighborhood structures across nodes, manifesting themselves in drastically different node degrees, give rise to the diverse behaviors of nodes and biased outcomes. In this paper, we first define and generalize the degree bias using a generalized definition of node degree as a manifestation and quantification of different multi-hop structures around different nodes. To address the bias in the context of node classification, we propose a novel GNN framework called Generalized Degree Fairness-centric Graph Neural Network (Deg-FairGNN). Specifically, in each GNN layer, we employ a learnable debiasing function to generate debiasing contexts, which modulate the layer-wise neighborhood aggregation to eliminate the degree bias originating from the diverse degrees among nodes. Extensive experiments on three benchmark datasets demonstrate the effectiveness of our model on both accuracy and fairness metrics.