Based on the theory of one-dimensional hexagonal piezoelectric quasicrystals, using complex function method, the anti-plane problem of asymmetrical cracks emanating from an elliptical hole in one-dimensional hexagonal piezoelectric quasicrystal bi-material was studied. Using conformal transformation, the crack problem on the edge of elliptical hole was transformed into the straight crack problem, combined with the Stroh formula, the analytical expressions of field intensity factors and energy release rates under the electrically impermeable and permeable conditions were obtained. Numerical example results show that the Griffith crack at the one-dimensional hexagonal piezoelectric quasicrystal interface is more likely to promote crack propagation than the cracks emanating from an elliptical hole. The more significant change in energy release rate takes into account the phason field, indicating that piezoelectric quasicrystal materials are more brittle than piezoelectric materials. The research provides a theoretical basis for the design and preparation of piezoelectric quasicrystal elements.