This paper presents a controller integrity monitoring method for a class of second-order nonlinear uncertain systems incorporating neural network-based adaptive control algorithms. The adaptive neural network model is employed to ensure robust tracking performance in the presence of certain modeling uncertainty under consideration during the offline controller design process. Based on Lyapunov stability analysis, an online controller integrity monitoring scheme is developed to detect the occurrence of controller software/algorithm faults and unanticipated physical component faults, which may lead to unstable learning behaviors and malfunctions of the adaptive controller. Adaptive thresholds for detecting controller malfunctions are derived, ensuring the robustness with respect to modeling uncertainty and neural network approximation error. Additionally, the detectability conditions are investigated, characterizing the class of detectable software faults and unanticipated physical faults. An upper bound on the fault detection time is also established. Simulation results are shown to illustrate the effectiveness of the proposed method.