In this paper, the finite-time tracking control problem of a class of multi-agent systems with nonaffine functions and uncertain nonlinearity is investigated, which is different from the existing on high-order multi-agent systems with pure feedback forms. The multi-agent systems considered in this paper, moreover, the nonaffine functions and uncertain nonlinearities are completely unknown, and the input of each follower agent is quantized through a hysteresis quantizer. Based on the help of the fuzzy logic systems approximator, an adaptive fuzzy finite-time tracking control protocol with adaptive update laws is presented by the backstepping technique. On the basis of the finite-time stability strategy and Bhat and Bernstein theorem, the finite-time stability of designed control protocol is fully analyzed. Under the proposed control protocol, it is indicated that the tracking error of each follower agent can press on a small neighborhood in a finite time. Finally, the effectiveness of designed control protocol in this paper is analyzed by numerical examples.