Theoretical investigations have been performed for the ground state ($S_0$) and the first excited state ($S_1$) of the hydrogen bonded green fluorescent protein (GFP) model. The potential energy surface (PESs) of $S_0$ was obtained by B3LYP method and that of $S_1$ was obtained by CIS method. Based on the relative stabilities of species and the energy barriers for the proton transfer, it was found that proton transfer could take place both under the ground state and the first excited state. As determined by the proton motions along the reaction coordinate, both the ground state proton transfer (GSPT) and the excited state proton transfer (ESPT) are considered as a concerted and asynchronous process.