This paper introduces the notion of quantum Müller automaton (LVMA), provides the concept of quantum recognizable finite step language and the means of quantum state construction, and then proves the fact that four types of LVMA can equivalently constructed from each other. By using those equivalent relations, it establishes the algebraic and level characterizations of quantum regular infinite languages, and also explores the closed properties of these quantum infinite languages in details under some infinite regular operations in particular at the same time. Meanwhile, this study shows that the behaviors of quantum Müller automata are precisely the quantum languages definable with sentences of the monadic second-order quantum logic (LVMSO), expanding the fundamental Büchi theorem to quantum setting.