<p>Quaternion tensors are currently applied widely across various scientific and engineering fields. However, the inherent multiple imaginary components of quaternions and their non-commutative multiplication pose significant challenges to the computational efficiency of quaternion tensors. In this paper, we introduce a complex representation matrix (CRM) for third-order quaternion tensors based on the specific algebraic structure. With the discrete Fourier transform, the novel CRM framework demonstrates distinctive advantages through its advantageous algebraic characteristics, which not only establish its theoretical uniqueness but also enable effective transformation of quaternion tensor optimization problems into the complex domain for efficient solution. The proposed CRM features a block-diagonal structure, enabling efficient large-scale computations and enhancing its suitability for high-dimensional problems. To evaluate its advantages in video inpainting, two CRM-based quaternion tensor completion methods are introduced. Simulation studies and example analyses confirm their effectiveness and efficiency.</p>