The boundary integral quadrature method (BIQM) has been successfully applied to the SH-wave scattering problem caused by a hill topography. In this paper, the BIQM is extended to simulate the SH-wave scattering problem by an orthotropic basin. Regarding the BIQM, the singular integral in the boundary integral equation can be regularized skillfully by introducing an adaptive exact solution. In addition, the calculation of the solid angle is also not required. For the orthotropic medium, the governing equation is not the standard 2D Helmholtz equation. The adaptive exact solution is thus re-derived to ensure that the governing equation and boundary continuity conditions are both satisfied. Consequently, the field values and their normal derivatives remain perfectly consistent with the original problem at each boundary collocation point. The field value is adaptively adjusted according to the position of the collocation point. After regularization, the remaining non-singular boundary integrals are described using a parametric representation of the boundary curve and are subsequently evaluated by using Gaussian quadrature. The boundary integral equation is transformed into a system of linear algebraic equations through the collocation method. Throughout this procedure, neither boundary element discretization nor interpolation functions for approximating boundary densities are required. The BIQM is also a meshfree method. Finally, the surface displacement amplitudes are calculated to verify the accuracy of the BIQM and to investigate the effects of different shear modulus ratios.