Since Thomason's paper in 1977 showing that the max-min powers of a fuzzy matrix either converge or oscillate with a finite period, many different algebraic operations are employed to explore the limiting behavior of powers of a fuzzy matrix, such as max-min/max-product/max-Archimedean t-norm/max-t-norm/max-arithmetic mean operations. In this article, we consider the max-quasi-arithmetic mean powers of a fuzzy matrix which is an extensive case of the max-arithmetic mean, max-root power mean and max-convex mean. We also show that the powers of such fuzzy matrices are always convergent. Some numerical examples are provided to show the situation of convergence.