In this paper, we address the consensus problem for nonlinear multi-agent systems with additive perturbations via a distributed model predictive control (DMPC) framework. To alleviate the computational and communication burdens, an event-triggered mechanism is proposed, which leverages error, its differentiation, and integration with an adaptive dynamic triggering threshold. Subsequently, a robust constraint is incorporated into the finite-horizon optimization problem to compensate for the effects of external additive disturbances. We rigorously prove that a minimum inter-event time exists, thereby guaranteeing Zeno-free behavior for the system even under external disturbances. Furthermore, the stability of the proposed algorithm is established under given conditions. Finally, a numerical simulation demonstrates that our scheme achieves multi-agent consensus and significantly reduces resource consumption without compromising control performance.