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Despite recent advances in achieving fair representations and predictions through regularization, adversarial debiasing, and contrastive learning in graph neural networks (GNNs), the working mechanism (i.e., message passing) behind GNNs inducing unfairness issue remains unknown. In this work, we theoretically and experimentally demonstrate that representative aggregation in message-passing schemes accumulates bias in node representation due to topology bias induced by graph topology. Thus, a \textsf{F}air \textsf{M}essage \textsf{P}assing (FMP) scheme is proposed to aggregate useful information from neighbors but minimize the effect of topology bias in a unified framework considering graph smoothness and fairness objectives. The proposed FMP is effective, transparent, and compatible with back-propagation training. An acceleration approach on gradient calculation is also adopted to improve algorithm efficiency. Experiments on node classification tasks demonstrate that the proposed FMP outperforms the state-of-the-art baselines in effectively and efficiently mitigating bias on three real-world datasets.