Nematic liquid crystals have been employed extensively in displays, and more recently in sensing technologies. In order to develop new applications for nematic-colloidal systems, a theoretical understanding of liquid crystal dynamics in the presence of colloidal particles is crucial. We first investigate the relaxation of two line defects of opposite strength in a confined nematic liquid crystal by solving the coupled tensor order parameter evolution and momentum balance equations in three dimensions. The inclusion of hydrodynamic interactions (HI) causes the defects to move at different velocities and slows the overall relaxation process, but these effects are suppressed by increasing the degree of confinement. The most notable flow features that develop in the system are large vortices that surround and shadow the defects as they approach each other. This treatment of HI is used to simulate the flow around colloidal particles. An anisotropic drag coefficient results, and the flows change the static position of the ring defects originally present around the particles' equators. These results can be used as a benchmark experiments ranging from particle diffusion to aggregation in nematic liquid crystals.