It mainly studies the least square solutions of quaternion Stein matrix equation. Firstly, by using the real representation method of quaternion matrix, the problem of solving quaternion matrix equation is transformed into the problem of solving corresponding real matrix equation. Secondly, according to the symmetric structural properties of the centrosymmetric (anti-centrosymmetric) matrix, using the $\mathcal{H}$-representation to extract independent elements and simplify the calculation, we give a new method for solving the least square centrosymmetric (anti-centrosymmetric) solution of the quaternion Stein matrix equation. Finally, the solution set of the least squares centrosymmetric (anti-centrosymmetric) solution of the equation and the necessary and sufficient conditions for the solution are given. The effectiveness of the method and results is demonstrated by numerical algorithms and examples.