For a class of nonlinear multi-agent systems, an adaptive dynamic surface state constrained quantized control scheme with an adjustable finite-time prescribed performance function is investigated. The major properties of the proposed control scheme are: 1) The adjustable finite-time prescribed performance function is combined into the barrier Lyapunov function to constrain the states of the multi-agent system, the function introduced can adjust its own parameters according to the current tracking error of the system without manual intervention. 2) By using the dynamic surface control method, the "differential explosion" phenomenon of the traditional backstepping control method is avoided, and the filtered compensating function is designed to eliminate the filter error and signal chattering caused by the dynamic surface method. 3) The RBF neural network is utilized to approximate the unknown nonlinear functions in the system, and the quantizer is introduced to reduce the communication burden of the system. The constructed quantization scheme can be realized only if the quantizer has the sector bounded property. The semi-globally uniformly bounded of all signals in the closed-loop system is demonstrated by stability analysis. Simulation section verifies the effectiveness of the proposed state constrained quantized control strategy.