The aim of this study is to propose a control system based on the model reference adaptive control methodology to solve a quadrotor trajectory tracking problem. As far as practical scenarios are concerned, the full dynamics of the quadrotor is usually unknown. So the plant model is considered, which includes uncertain nonlinear terms caused by the aerodynamic friction, blade flapping and the fact that quadrotor mass and inertia moments can vary from their nominal values. Using it in conjunction with the differential flatness approach, the explicit equation of the parameter uncertainty for the position control loop is obtained. Having analyzed it, the neural network (NN) is chosen as a compensator, as well as the set of NN input signals is justified. The parameters of such network are adjusted using the derived adaptive laws. Assuming perfect attitude loop tracking, which is provided by the system previously proposed by the authors, position controller guarantees asymptotic stability of the tracking error. Numerical experiments results verify the effectiveness of the proposed method and show to what extent the parameter uncertainty has been compensated.