This article addresses the distributed consensus problem of multi-agent systems with Lur'e nonlinear dynamics under directed graphs based on event-triggered strategy. With the state-dependent event-triggered thresholds, a novel distributed event-triggered consensus controller is designed, under which the consensus can be achieved with zero final consensus error. The proposed strategy has several distinguishing features, including the fact that individual agent does not require continuous, or even periodic, communication with their neighbours to update the controller or monitor the triggering condition, and all the required parameters can be locally determined by the agent. Furthermore, we also prove that there is no Zeno behaviour existed. Finally, the simulation example about Chua's circuit is given to illustrate the theoretical analysis.