High-order quantum correlation provides powerful methods to reveal the quantum many-body behavior of ultracold atomic gases. In this work, the second-order quantum correlation is adopted to study the many-body behavior of ultracold Bose gases in the presence of both a two-dimensional optical lattice and weak disorder. According to investigations, it is found that even a weak disorder plays a significant role in the quantum many-body behavior, which manifests itself through the second-order quantum correlation. With the Bogoliubov theory, our studies show that both interatomic interactions and weak disorder would destroy the first-order quantum coherence of the condensate because of the depletion, and the resulting depletion has significant characteristic in the second-order correlation of the system.