This paper studies fully distributed formation control problems for multi-quadrotor systems with external disturbances via an event-triggered approach. For leader-follower multi-quadrotor systems, a fully distributed event-triggered formation control (DETFC) framework is proposed by utilizing an adaptive sliding-mode control approach, which does not rely on any global information of the network topology. The finite-time reachability of the sliding-mode surface can be guaranteed for the states of the nonlinear, coupled and underactuated system subject to external disturbances. Meanwhile, the zeno behavior of the proposed event-triggered protocol is proved to be excluded. A novel dynamic sliding-mode surface is designed to guarantee the formation performance as the quadrotor state trajectories move on the constructed sliding manifold. Via Lyapunov stability theory, sufficient conditions to ensure the formation results are derived for multi-quadrotor systems. Simulations and experiments are conducted to validate the effectiveness of the proposed control scheme.