We 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 causes the defects to move at different velocities and slows the overall relaxation process, but this effect is suppressed by increasing the degree of confinement. The most notable flow features that develop in the system are large vortices that surround and follow the defects as they approach each other.