Material Point Method (MPM) demonstrates excellent capability in simulating large deformations by circumventing the mesh distortion issues inherent in the Finite Element Method (FEM). However, MPM suffers from cell-crossing error and volumetric locking when modeling large deformations of near-incompressible materials, such as rubbers and biological tissues. A unified exact quadrature based on the particle domain is introduced into the B-spline Material Point Method (uqBSMPM) to mitigate non-physical pressure oscillations caused by cell-crossing errors and volumetric locking in traditional MPM. The high-order continuity of B-spline basis functions enables the implementation of the proposed exact quadrature over the material particle's domain in a unified analytical form, free from singularities at the background grid nodes. Numerical results demonstrate that the computational accuracy of the proposed method is comparable to that of the explicit dynamic finite element method for the linear elastic small deformation problem. With the elimination of pressure oscillations, the large deformation of near-incompressible materials under shear and torsion has been accurately modeled, suggesting the uqBSMPM's ability to model practical soft material applications.