This study describes a novel approach utilizing quaternions for loop closure constraints in rigid origami structures. Our research outlines the modeling of rotation and loop closure constraint with quaternions, emphasizing their advantages over traditional matrix-based methods. We also present numerical solutions using the Newton-Raphson method for finding folding angles in origami patterns, comparing the performance of quaternion-based and matrix-based methods through examples such as the Miura-ori pattern and a 4-closed-vertex pattern. Results indicate the effectiveness of the quaternion approach, particularly in multi-vertex origami problems.