In this paper we propose the Hamiltonians of the generalized SU_q(1|2) Gaudin model corresponding to the periodic generalized t-J model. With the help of the well defined graded quantum determinant, we obtain the eigenstate and eigenvalues of the generating function and the Hamiltonians of the Gaudin model in the fermionic background in the framework of the graded quantum inverse scattering method. The Bethe ansatz equations are also obtained.