To design and prepare high-quality one-dimensional hexagonal quasicrystal nano-composites, the interface and interface phase models were applied to study the infinite one-dimensional hexagonal quasicrystal anti-plane fracture problem with cylindrical inclusions containing nano coatings by using the complex function method and Gurtin-Murdoch's surface/interface elasticity theory. Under two different models, the series form expressions of phonon and phason field stress fields in matrix, coating and inclusion were obtained, respectively. Numerical examples were used to analyze the effects of interface elastic constants and size effects on the stress field around inclusions. The results showed that the positive or negative values of interface elastic constants would affect the stress field around nano-inclusions. As the size of nano-inclusions increased, the stress field exhibited significant size dependence, and surface effects had significant differences in their effects on the stress fields of dimensionless phonon and phason fields. The relevant results provide a certain theoretical reference for studying the mechanical behavior of quasicrystalline nano-inclusions.