To investigate the special electrical phenomena in a graphene/ferroelectric field-effect transistor, we perform density-functional first-principles calculations on a graphene/LiNbO3 (0001) hybrid structure, and then study the effects of a Au counter electrode and water adsorption. Graphene is p-type and n-type doped by a LiNbO3 positive and negative surface, respectively, and the conductivity increases while the Dirac cone persists. The Au counter electrode introduces an additional charge injection relative to the surface polarity and weakens the electrostatic doping effect. With the help of the electrons from the Au counter electrode, water adsorption could switch the conductive type of graphene. We also find an interesting phenomenon whereby the coupling of graphene with a negative surface is significantly stronger than that with a positive surface, possibly related to the different polar terminals (LiO3/Nb).