Firstly, the extended Stroh formulism is used to study the Green's function of an infinite octagonal two-dimensional quasicrystals plate with an elliptical hole. And the fundamental solution of the boundary element method for solving the finite octagonal two-dimensional quasicrystals plate with an elliptical hole is obtained. Secondly, the boundary integral equation is established by using the weighted residual method. The boundary integral equation with unknown quantity is discretized by linear interpolation function and Gauss integral, and the discrete format is reorganized to form a linear system of equations with unified variables. Finally, the hole edge stress of the elliptical hole is numerically solved, and the numerical results of the finite plate are compared with the analytical solution of the infinite plate to verify the effectiveness of the boundary element method. The influence of the size of the plate, the size of the hole and the inclination angle on the stress at the edge of the hole under vertical tension are further analyzed. The results of numerical examples show that the stress concentration at the edge of the hole is more obvious with the increase of the size of the elliptical hole. The inclination of the elliptical hole aggravates the stress concentration at the edge of the hole. The stress intensity factor at the crack tip increases with the growth of the crack.