Numerical simulation of complex geometries can be an expensive and time-consuming undertaking, in particular, due to the lengthy preparation of geometry for meshing and the meshing process. This problem becomes more apparent in cases of large deformation problems, where intermediate solution steps are necessary to achieve convergence. Various techniques were suggested to tackle this problem, including extended finite element, meshless, Fourier transform, and immersed boundary methods. Immersed boundary methods rely on embedding the physical domain into a Cartesian grid of finite elements and resolving the geometry only by adaptive numerical integration schemes. However, the accuracy, robustness, and efficiency of immersed or cut cell approaches depend crucially on the integration technique applied to cut cells. In this work, we utilize an innovative algorithm for boundary-conformal quadrature that relies on a high-order reparameterization of trimmed elements to address elasticity problems. We accomplish this using spline-based immersed isogeometric analysis, which eliminates the need for body conformal finite element mesh. The Gauss points on trimmed elements are obtained through a NURBS reparameterization of the physical subdomains of the Cartesian grid to ensure precision integration with minimal quadrature points. The 2D plate with hole problem serves as a benchmark problem for comparing the algorithm with conformal isogeometric analysis and the finite cell method. The results demonstrate that the adopted boundary conformal immersed Isogeometric analysis converges with optimal rates, thus demonstrating the efficiency and the precision of the method.