Localized plasmon modes are excited and probed in a large-area grating-gate GaN/AlGaN high-electron-mobility transistor structure embedded in a Fabry-Perot cavity using a terahertz time-domain spectroscopy (THz-TDS) at cryogenic temperature. Determined by the length of grating finger and the electron concentration, the frequency of localized plasmon modes can be continuously tuned by the gate voltage in the spectral range from 0.1 THz to 1.5 THz. When the plasmon frequency is tuned to be in resonance with the terahertz Fabry-Perot cavity mode, a strong coupling between the plasmon mode and the cavity mode is observed and the terahertz plasmon-polaritons are formed in such a cavity-coupled two-dimensional electron system. The electromagnetic simulations have confirmed the strong coupling between them.