<p>The quadratic matrix equations have widespread applications in scientific computing and engineering domains. This paper investigates a class of quadratic matrix equations arising in noisy Markov chains. Firstly, the existence of the M-matrix solution is proved in a general case. Then, a structure-preserving doubling algorithm is constructed to solve the equation, and a convergence analysis of the algorithm is given. Finally, numerical examples are given to show that the structure-preserving doubling algorithm presented in this paper is feasible and is more effective than some methods under certain conditions.</p>