Real-time damage detection algorithms deployed on Unmanned Aerial Vehicles (UAVs) can support flight control in real time, enabling the capture of higher quality inspection data. However, three challenges have hindered their wider application: 1) Existing anchor-based damage detectors cannot generalize well to real-world scenarios and degrade the detection speed; 2) Prior studies exhibit a low detection accuracy; 3) No previous study considers the energy consumption issue of the damage detector, limiting the UAVs' flight time. To meet these challenges, this paper presented the YOLOv6s-GRE-quantized method, which is an energy-efficient anchor free and real-time damage detection method built on top of the YOLOv6s algorithm. Firstly, the YOLOv6s-GRE method was presented, where a generalized feature pyramid network (GFPN), a reparameterization efficient layer aggregation network (RepELAN) and an efficient detection head were introduced into the YOLOv6s. Comparison experiments showed that the YOLOv6s-GRE method, in contrast to YOLOv6s, advanced 2.3 percentage points in the metric of mAP50, while maintaining comparable detection speed and without requiring an increase in model size. The YOLOv6s-GRE model was then reconstructed by the RepOptimizer (RepOpt) to equivalently transform the YOLOv6s-GRE into a quantization-friendly model for addressing the quantization difficulty of the reparameterization model. Finally, the YOLOv6s-GRE model with RepOpt was quantized by the partial quantization-aware training technique, expediting the detection speed by 83.5% and saving energy by 79.7% while still maintaining a comparable level of detection accuracy. Implementing of this proposed method can significantly boost bridge inspection productivity.