Due to the physical nature of color images, color image processing such as denoising and inpainting has shown extensive and versatile possibilities over grayscale image processing. The monochromatic and the concatenation model have been widely used to process color images by processing each color channel independently or concatenating three color channels as one unified one and then used existing grayscale image processing methods directly without specific operations. These above schemes, however, have some limitations: (1) they would destroy the inherent correlation among three color channels since they cannot represent color images holistically; (2) they usually focus on one specific handcrafted prior such as smoothness, low-rankness, or even deep prior and thus failing to fuse deep and handcrafted priors of color images flexibly. To conquer these limitations, we propose one unified model to integrate deep prior and low-rank quaternion prior (DLRQP) for color image processing under the plug-and-play (PnP) framework. Specifically, the quaternion representation with low-rank constraint is introduced to denote the color image in a holistic way and one advanced denoiser is adopted to explore the deep prior in an iterative process. To tightly approximate the quaternion rank, one nonconvex penalty function is further utilized. We derive an alternate iterative approach to tackle the proposed model. We empirically demonstrate that our model can achieve superior performance over existing methods on both color image denoising and inpainting tasks.