<p>Based on the surface elasticity model and complex variable function theory, this study investigates the model III fracture problem of a nano-lip shaped hole with four cracks in one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) by constructing a new conformal mapping. The analytical solution field for intensity factors and energy release rate (ERR) were obtained. By degenerating the relevant parameters, results for some classical defects can be derived. Numerical examples were then used to dynamic analyze the effects of the hole size, crack length, external mechanical, electrical loads, and phonon-phason ASE coupling coefficient on the fracture mechanical behavior. The results show that when defects reach nano sizes, surface effects are generated under the conditions of mutual coupling between the phonon field, phason field, and electric field. The smaller the defect size, the more pronounced the surface effect. As the defect size increases, the impact of surface effects on fracture behavior gradually diminishes, eventually converging to the results of classical fracture theory. The findings of this study can offer theoretical guidance for the structural design of nano-quasicrystal materials and the fracture mechanics research of nanoscale defects, providing a strong theoretical foundation for the development and utilization of engineered materials.</p>