A derivation of the vibrational Hamiltonian in generalized (internal) coordinates within a grid representation using the Fourier Grid Hamiltonian (FGH) method is presented. The objective of the theory is to make possible the solution of vibrational problems in two or more dimensions in terms of internal coordinates. These coordinates are often the ones of choice when the vibrations of interest are localized or when only selected coordinates are considered in a larger system. As in the case of the original FGH method, the matrix elements are easy to evaluate in a fast and robust manner. The method is tested on two different molecular systems, FHF- and picolinic acid N-oxide, both containing strong hydrogen bonds. The illustrative problems are two-dimensional and are highly anharmonic. The importance of both the coupling terms, as well as the variable reduced masses, required by the formulation of the Hamiltonian in generalized coordinates, are examined, and the formalism is shown to be robust in that identical results are obtained using different sets of internal coordinates applied to the same physical problem. Good agreement between calculated and observed vibrational frequencies is also obtained.