Two-dimension discrete Fourier transform (2-D DFT) is a fundamental tool in grays-scale image processing. In color imaging, this transform is used to process separately color channels and such processing does not consider interactions between the color channels. The concept of the quaternion discrete Fourier transform (QDFT) became a very popular topic in color imaging. The color image from one of the color model, for instance the RGB model, can be transformed into the quaternion algebra and be represented as one quaternion image which allows to process simultaneously of all color components of the image. In this work, we describe the algorithm for the 2-D left-side QDFT which is based on the concept of the tensor representation when the color or quaternion image is described by a set of 1-D quaternion signals and the 1-D left-side QDFTs over these signals determine values of the 2-D left-side QDFT at corresponding subset of frequency-points. The efficiency of the tensor algorithm for calculating the fast left-side 2-D QDFT is described and compared with the existent methods.