The difference between quantum plasma and classical plasma is mainly reflected in the following two aspects: 1) the statistical equilibrium state of the system changes from the classical Maxwell distribution to the Fermi-Dirac distribution; 2) the single-particle quantum wave effect of electrons cannot be avoided. Corresponding to the Vlasov equation in the classical plasma, the kinetic equation of the quantum plasma is the Wigner equation, but the numerical solution of the Wigner equation is more complicated than the Vlasov equation. In this paper, we propose a new method based on flux balance and Fourier spectrum methods. The hybrid method is used to solve the Wigner-Poisson equations. This method adopts different time advancing algorithms in the coordinate and velocity spaces. Compared with the general discrete Euler method, it can significantly improve the accuracy of nonlinear simulation results. This paper investigates the behavioral changes of some common electrostatic kinetic instabilities in quantum plasmas through this method; verifies the reliability of the code through linear eigensolutions, and then simulates some nonlinear phenomena, including Nonlinear Landau damping and nonlinear saturation for two-stream instability, etc.