Generating realistic human flows across regions is essential for our understanding of urban structures and population activity patterns, enabling important applications in the fields of urban planning and management. However, a notable shortcoming of most existing mobility generation methodologies is neglect of prediction fairness, which can result in underestimation of mobility flows across regions with vulnerable population groups, potentially resulting in inequitable resource distribution and infrastructure development. To overcome this limitation, our study presents a novel, fairness-aware deep learning model, FairMobi-Net, for inter-region human flow prediction. The FairMobi-Net model uniquely incorporates fairness loss into the loss function and employs a hybrid approach, merging binary classification and numerical regression techniques for human flow prediction. We validate the FairMobi-Net model using comprehensive human mobility datasets from four U.S. cities, predicting human flow at the census-tract level. Our findings reveal that the FairMobi-Net model outperforms state-of-the-art models (such as the DeepGravity model) in producing more accurate and equitable human flow predictions across a variety of region pairs, regardless of regional income differences. The model maintains a high degree of accuracy consistently across diverse regions, addressing the previous fairness concern. Further analysis of feature importance elucidates the impact of physical distances and road network structures on human flows across regions. With fairness as its touchstone, the model and results provide researchers and practitioners across the fields of urban sciences, transportation engineering, and computing with an effective tool for accurate generation of human mobility flows across regions.