By the discrete position representation method,the vibrational-rotational levels of some bound-state and the Franck-Condon factors of rovibrational transitions in non-adiabatic approximations for diatomic molecules are calculated. The results are in accord with the literature values very well. The relative errors are all smaller than 1%. These results may be useful for the experimental study of rovibrational spectra of the diatomic molecules. The computational procedure can be popularized to the numerical calculations for other diatomic molecules and the multi-dimensions quantum systems.