The rapid growth of the Unmanned Aerial Vehicle industry has increased the demand for robust control systems for commercial quadcopters, whose dynamic models are often unknown, to ensure reliable performance under adverse conditions. In this context, this paper proposes Deep Neural Network-Based LQR Adaptive Control (DNN-LQR-AC), an Adaptive Control (AC) strategy implemented using the Robot Operating System (ROS) software framework. DNN-LQR-AC relies on a simplified linearized dynamic model and combines an LQR controller with an adaptive term, which is updated in real-time using Deep Neural Networks (DNNs). The proposed solution also features cubic spline-based trajectory generation to provide continuous reference trajectories for the controller. Experimental validation using a Hardware-in-the-Loop approach on a DJI Matrice 100 quadcopter demonstrates that DNN-LQR-AC outperforms PID, LQR, and LQR-AC controllers, achieving superior position control in trajectory tracking under varied wind conditions, highlighting its applicability in real-world scenarios.