<p>This paper proposes novel approaches for low-rank approximation of quaternion tensors. The first method employs quasi-norms to approximate low-rank structure using the QT-product, a generalization of the L-product to N-mode quaternions. The second method leverages non-convex norms to approximate both Tucker and TT-ranks for tensor completion. We demonstrate that the proposed models yield more accurate approximations compared to traditional convex relaxations, such as the nuclear norm. Furthermore, we establish theoretical convergence guarantees to support their effectiveness. Extensive numerical experiments confirm the superiority of our methods in tensor completion and denoising tasks. These results validate the use of non-convex surrogate functions in quaternion tensor models for robust and accurate high-dimensional data reconstruction.</p>