Without the limitations of labelled distorted images for training, the unsupervised blind colour image quality assessment (UBCIQA) model shows excellent generalization performance. However, existing UBCIQA methods fail to consider the correlation between different colour channels. This paper proposes an efficient UQBCIA algorithm. Placing R, G, and B channels of a colour image into the three imaginary parts of a quaternion, the proposed UQBCIA obtains the quaternion representation of a colour image, which allows UQBCIA to process the colour image as a whole while keeping its three colour channels dependent. The naturalness, structural and texture statistic features are then extracted to fit a multivariate Gaussian (MVG) model. The quality of the test colour image is calculated as the MVG distance between the fitted pristine images and the test image. The proposed method is evaluated from three aspects: (1) prediction accuracy, (2) computational complexity and (3) robustness on five colour databases. The results demonstrate that the proposed UQBCIA algorithm outperforms state-of-the-art unsupervised and classical supervised BIQA methods.