Quaternion moments are widely used in several applications, such as image classification, object recognition, and multimedia security. The computation of these moments requires a vast computational time, especially for big-size images. Several attempts to accelerate quaternion moments are not enough to process big-size color images with the desired speedup. In this work, we proposed a new parallel framework for fast computation of quaternion moments in Cartesian coordinates using multi-core CPUs and many-core graphics processing units (GPUs) with the Compute Unified Device Architecture (CUDA). We called the proposed unified computational framework “CUDAQuat.” This framework was tested by eleven sets of quaternion moments. Several applications executed using the proposed parallel framework where the CPU times, execution-time-improvement ratio, and speedup were reported. The evaluation outlined significant speedup over the single-core CPU implementation, where the proposed framework accelerated several sets of quaternion moments with speedup 600x.