In future 6G communication systems, large-scale antenna arrays promise enhanced signal strength and spatial resolution, but they also increase the complexity of beam training. Moreover, as antenna counts grow and carrier wavelengths shrink, the channel model transits from far-field (FF) planar waves to near-field (NF) spherical waves, further complicating the beam training process. This paper focuses on millimeter-wave (mmWave) systems equipped with large-scale uniform planar arrays (UPAs), which produce 3D beam patterns and introduce additional challenges for NF beam training. Existing methods primarily rely on either FF steering or NF focusing codewords, both of which are highly sensitive to mismatches in user equipment (UE) location, leading to high sensitivity to even slight mismatch and excessive training overhead. In contrast, we introduce a novel beam training approach leveraging the beam-diverging effect, which enables adjustable wide-beam coverage using only a single radio frequency (RF) chain. Specifically, we first analyze the spatial characteristics of this effect in UPA systems and leverage them to construct hierarchical codebooks for coarse UE localization. Then, we develop a 3D sampling mechanism to build an NF refinement codebook for precise beam training. Numerical results demonstrate that the proposed algorithm achieves superior beam training performance while maintaining low training overhead.