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 itself. To tackle this problem, immersed boundary and fictitious domain 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 depends crucially on the integration technique applied on trimmed cells. This issue becomes more apparent in nonlinear problems, where intermediate solution steps are necessary to achieve convergence. In this work, we adopt an innovative algorithm for boundary conformal quadrature that relies on a high-order B-spline re-parameterization of trimmed elements to address small and large deformation elasticity problems. We accomplish this using spline-based immersed isogeometric analysis, which eliminates the need for body conformal finite element mesh. The integration points are obtained by applying classical Gauss quadrature to conformal re-parameterizations of the cut elements, whereas the discretization itself is not refined. This ensures a precise integration with minimum quadrature points and degrees of freedom. The proposed immersed isogeometric analysis with boundary conformal quadrature is evaluated on benchmark problems for 2D linear and nonlinear elasticity. The results show convergence with optimal rates in h-and k-refinement, thus demonstrating the efficiency and the precision of the method. As demonstrated, in conjunction with the simple to implement penalization and deformation map resetting approaches in the fictitious domain, it performs robustly also for finite deformations. Furthermore, it is exemplified that the method can be easily applied for multiscale homogenization of microstructured materials in the large deformation regime.