Since the quantum field theory was invented at the end of 1920s, attempts have been made to apply it to gravitational field. After more than 20 years, the formal quantum theory of gravity, which describes any systems as an action function in a canonical Hamiltonian method, achieved for the first time a state of completion. Contrary to the situation held for the canonical theory, a covariant treatment also was developed to deal with the physical conditions such that effects of vacuum processes must be taken into account. In this review, we start from a brief introduction to the history of construction of quantum theory of gravity and consider the applications of two quantization methods to inspiraling neutron star binary systems. We mainly focus on the gravitational quantum effects on dynamics of the gravitational quanta radiated from different systems by using canonical method and on the gravitational potential of the binaries by employing the covariant treatment. The possible detections and constraints to the quantum effects are also discussed.