Quadrotors possess traits such as under-actuation, nonlinearity, and strong coupling. Quaternions are primarily used for attitude calculations in drones, with error quaternions seldom being employed directly in the control of specific quadcopter drones. This paper focuses on the low tracking accuracy and weak anti-interference ability of quadcopter drones in trajectory-tracking control. By establishing the quadcopter quaternion model, a controller based on quaternion error is designed through a combination of fractional-order PID control with S-plane control. Trajectory-tracking experiments demonstrate that, in comparison with fractional-order PID, this method exhibits strong wind disturbance resistance and high tracking accuracy.