Constrained solutions are required in some practical problems. This paper studies the problem of different constrained solutions for a class of multivariate quadratic matrix equations, which has rarely been studied before. First, the quadratic matrix equations are transformed into linear matrix equations by Newton's method. Second, the different constrained solutions of the linear matrix equations are solved by the modified conjugate gradient method, and then the different constrained solutions of the multivariate quadratic matrix equations are obtained. Finally, the convergence of the algorithm is proved. The algorithm can be well performed on the computers, and the effectiveness of the method is verified by numerical examples.