Since the magnitudes of the Goos-H?nchen (GH) shifts in multilayered photonic crystals are generally small, it is desirable to find the alternative configurations to achieve the large GH shift. In this work, we investigated the GH shift of the reflected wave in the structure with a metal layer, a dielectric material, and the quasiperiodic photonic crystal by the transfer matrix method, where the quasiperiodic photonic crystal is composed of a dielectric material and monolayer graphene arranged in a Fibonacci sequence. It is found that the GH shift can be enhanced up to 7330 times of the incident wavelength at the specified operating wavelength 2 μm due to the excitation of surface plasmon polaritons of metal. In addition, we discussed the influence of the optical parameters of monolayer graphene, and the thickness of the dielectric material on the GH shift, and confirmed that changing these parameters could achieve the control of GH shift.