In this paper, a fully-connected (FC) hybrid precoding architecture in the Terahertz (THz) band is investigated for 6G wireless communications. The approximated spectral efficiency of the THz FC hybrid precoding architecture is derived. Inspired by this, a low-complexity column-by-column (CBC) algorithm is proposed for the THz FC hybrid precoding. Simulation results show that, on the one hand, the spectral efficiency of the CBC algorithm approaches the fully digital precoding scheme and outperforms the existing algorithm with a low complexity. On the other hand, the approximated spectral efficiency of the FC hybrid precoding is accurate. Furthermore, some open problems of the Terahertz hybrid precoding are presented.