This paper proposes hybrid beamforming designs for millimeter-wave (mmWave) multiple-input multiple-output (MIMO) backhaul systems equipped with uniform planar arrays (UPAs) of dual-polarization antennas at both the transmit and receive base stations. The proposed beamforming designs are to near-optimally solve optimization problems taking the dual-polarization UPA structure into account. Based on the solutions of optimization problems, this paper shows it is possible to generate the optimal dual-polarization beamformer from the optimal single-polarization beamformer sharing the same optimality. As specific examples, squared error and magnitude of inner product are considered respectively for optimization criteria. To optimize proposed beamformers, partial channel information is needed, and the use of low overhead pilot sequences is also proposed to figure out the required information. Simulation results verify that the resulting beamformers have the most uniform gain (with the squared error criterion) or the highest average gain (with the magnitude of inner product criterion) in the covering region with the UPA of dual-polarization antennas.