In this paper, the adaptive neural time-varying full-state constraints quantized consensus control is investigated for a class of nonlinear multiagent network systems with switching topologies. In most existing state constraints control schemes, the state constraints are achieved based on error constraints, and the feasibility conditions are required, which is a difficult problem for the application of controllers. Thus, based on the adaptive neural control method, this paper designs the integral Lyapunov functions to remove the restriction of the feasibility conditions completely. Therefore, the boundaries of state constraints control can be designed directly without feasibility conditions. Moreover, the boundaries of state constraints are considered as time-varying since there exist various factors affecting the boundaries, which is more suitable for practical multiagent systems. In multiagent network control, data transmission is a crucial element for system performance. For this problem, this paper designs the time-varying state constraints input quantization signals, which can reduce chattering and achieve low communication rates in the digital signal transmission for the multiagent systems under switching topologies communication networks. Then, the proposed strategy is validated in the simulation section. All the closed-loop signals are bounded under the proposed strategy.