Graph neural networks (GNNs) have emerged as the mainstream paradigm for graph representation learning due to their effective message aggregation. However, this advantage also amplifies biases inherent in graph topology, raising fairness concerns. Existing fairness-aware GNNs provide satisfactory performance on fairness metrics such as Statistical Parity and Equal Opportunity while maintaining acceptable accuracy trade-offs. Unfortunately, we observe that this pursuit of fairness metrics neglects the GNN's ability to predict negative labels, which renders their predictions with extremely high False Positive Rates (FPR), resulting in negative effects in high-risk scenarios. To this end, we advocate that classification performance should be carefully calibrated while improving fairness, rather than simply constraining accuracy loss. Furthermore, we propose Fair GNN via Structural Entropy (\textbf{FairGSE}), a novel framework that maximizes two-dimensional structural entropy (2D-SE) to improve fairness without neglecting false positives. Experiments on several real-world datasets show FairGSE reduces FPR by 39\% vs. state-of-the-art fairness-aware GNNs, with comparable fairness improvement.