<p>The Moore-Penrose inverse of quaternion matrices efficiently handles the underdetermined, overdetermined, and singular quaternion linear systems. In this paper, we propose two new iterative algorithms for computing the Moore-Penrose inverse of arbitrary quaternion matrices. Specifically, by utilizing certain properties of the Moore-Penrose inverse, we transform the problem of computing the Moore-Penrose inverse of quaternion matrices into solving a class of quaternion matrix equations. Subsequently, using the modified conjugate gradient method, we obtain an approximate solution to the Moore-Penrose inverse of quaternion matrices. Finally, we demonstrate the efficiency of the proposed methods through numerical examples and verify the results drawn in this paper.</p>