This paper proposes a novel method for designing compactly supported biorthogonal graph wavelet filter banks with flat spectral responses. We firstly construct a class of biorthogonal graph filter banks by using the polynomial half-band kernels, and then present a design method for the polynomial half-band kernel. The proposed design method utilizes the PBP (Parametric Bernstein Polynomial), which ensures that the half-band kernel has the specified zeros at A = 2, that is, the flatness constraints both in passband and stopband. Therefore, the resulting graph filters have the flat spectral responses. Furthermore, we apply the Remez exchange algorithm to minimize the spectral error by using the remaining degree of freedom. Finally, two design examples are given to demonstrate the effectiveness of the design method proposed in this paper.