This paper is concerned with the problem of mean square consensus for nonlinear multi-agent systems with state-dependent noise perturbations. A distributed event-triggered mechanism which can be used to reduce the number of controller updates and communication load is developed to apply in stochastic dynamical systems. By combining the tools of graph theory and stochastic analysis, the sufficient conditions are given to ensure that mean square consensus in the multi-agent systems can be achieved exponentially. The convergence rate is also analytically derived. Moreover, the feasibility of the proposed distributed event-triggered strategy is further verified by exclusion of Zeno behaviour. Finally, some numerical examples are given to demonstrate the effectiveness of the obtained results.