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Federated learning (FL) allows UAVs to collaboratively train a globally shared machine learning model while locally preserving their private data. Recently, the FL in edge-aided unmanned aerial vehicle (UAV) networks has drawn an upsurge of research interest due to a bursting increase in heterogeneous data acquired by UAVs and the need to build the global model with privacy; however, a critical issue is how to deal with the non-independent and identically distributed (non-i.i.d.) nature of heterogeneous data while ensuring the convergence of learning. To effectively address this challenging issue, this paper proposes a novel and high-performing FL scheme, namely, the hierarchical FL algorithm, for the edge-aided UAV network, which exploits the edge servers located in base stations as intermediate aggregators with employing commonly shared data. Experiment results demonstrate that the proposed hierarchical FL algorithm outperforms several baseline FL algorithms and exhibits better convergence behavior.