Based on the fundamental equations of piezoelasticity of quasicrystal media, using the symmetry operations of point groups, the linear piezoelasticity behavior of one-dimensional (1D) hexagonal quasicrystals is investigated and the piezoelasticity problem of 1D hexagonal quasicrystals is decomposed into two uncoupled problems, i.e., the classical plane elasticity problem of conventional hexagonal crystals and the phonon–phason-electric coupling elasticity problem of 1D hexagonal quasicrystals. The final governing equations are derived for the phonon–phason-electric coupling anti-plane elasticity of 1D hexagonal quasicrystals. The complex variable method for an anti-plane elliptical cavity in 1D hexagonal piezoelectric quasicrystals is proposed and the exact solutions of complex potential functions, the stresses and displacements of the phonon and the phason fields, the electric displacements and the electric potential are obtained explicitly. Reducing the cavity into a crack, the explicit solutions in closed forms of electro–elastic fields, the field intensity factors and the energy release rate near the crack tip are derived.