With the enlargement of antenna apertures in 6G Terahertz (THz) communications, the Rayleigh distance expands significantly, rendering near-field propagation a dominant scenario in THz links. Beyond conventional Line-of-Sight (LoS) and Non-Line-of-Sight (NLoS) conditions, quasi-LoS scenarios with partial obstructions have emerged as a critical challenge. Airy beams offer a promising solution to circumvent obstacles due to their unique curving trajectory. However, existing Airy beam training methods typically rely on parameter-based sampling or exhaustive search, leading to significant pilot overhead and low training efficiency. In this paper, an efficient Airy beam training framework is proposed to address this research gap. First, the theoretical bounds of Airy beam generation under finite apertures to prune physically invalid codewords are derived. Based on this, a two-stage Non-Uniform Polar Codebook (NUPC) design is presented, utilizing a probing mechanism to resolve the bending direction and a polar-domain spatial sampling strategy to generate Airy beams. To address ultra-low latency requirements, a Fast-Scanning 1D Codebook (FS1C) is further developed that sweeps the entire LoS region with minimal codewords. Simulation results demonstrate that NUPC achieves a higher average spectral efficiency (SE) by 13.4 bit/s/Hz while reducing training overhead by 54.2% compared to the state-of-the-art hierarchical focusing-Airy codebook (HFAC). Furthermore, FS1C reduces overhead by 92.9% with only a marginal 0.3 bit/s/Hz reduction compared with HFAC.