Quadrotors are expected to play an active role in out-of-sight areas such as equipment inspection and transportation of supplies to disaster areas. However, ensuring stable and highly accurate automatic flights has become a significant challenge. Trajectory tracking control methods that combine adaptive output feedback control based on almost strictly positive real characteristics and backstepping strategy have been proposed for a quadrotor control system with nonlinearities. These methods have simple structures and are robust. However, existing adaptive control methods have the problem of singularities, owing to the use of Euler angles in the modeling of quadrotors. In this paper, we propose an adaptive control method that enables a wider range of attitude changes using quaternions. The effectiveness of the proposed method is validated through numerical simulations.