Emerging HDR videos enable the recording of adequate luminance information and representing realistic scenes to the audience. Due to the high precision of the HDR data recorded in the floating-point format, a quantization process is required to convert HDR data to integer data for compatibility with current transmission and display systems. In this study, a novel attention-aware quantization method is presented that attempts to preserve the contrast details in the region of interest of the human visual system. This method was applied in the context of HDR video coding. The proposed coding solution was compared with the current anchor solution in terms of the quality of the reconstructed video. Experimental results show that the proposed solution is able to improve the visual quality of encoded video with respect to the anchor solution. Additionally, the proposed solution achieves a bit-rate gain over the anchor with reference to the objective evaluation results.