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Monolayer MoS2 has attracted wide attention because of its finite bandgap, and it has become a potential candidate for the investigation of the Goos–Hänchen (GH) shift. However, the magnitude of the GH shift in free-standing monolayer MoS2 is small, which greatly hinders its possible applications in the photoelectric sensors and detectors. We have theoretically designed a defective quasiperiodic photonic crystal and investigated its GH shift, where monolayer MoS2 is sandwiched between two quasiperiodic photonic crystals arranged by the Pell sequence. By optimizing the thicknesses of all the components and the period number of the Pell quasiperiodic photonic crystal, we find that the GH shift of the designed structure is significantly enhanced at the specific working wavelength. In addition, we discuss the influence of the thicknesses of the dielectric components on the GH shift. Our work confirms that the quasiperiodic photonic crystal structure has the ability to enhance the GH shift of monolayer transition metal dichalcogenides, which provides a new platform for the GH investigations and greatly promotes the applications of this defective structure in optoelectric devices.