The coherently coupled nonlinear Schrdinger (NLS) equation of the propagation of light pulse in fiber Bragg gratings has been utilized to study the modulation instability of laser pluses in fiber Bragg gratings with Gaussian apodization function, not only in the anomalous dispersion regime, but also in the normal dispersion regime. The results show that modulation instability can occur in both regimes. The regular gain spectra are observed obviously when input power reaches a definite value in the anomalous dispersion regime. The modulation instability can increase from a fixed value to infinity when the input power is in the region of modulation instability in normal dispersion regime. Furthermore, the gain spectra are restrained by Gaussian apodization function in both regimes.