Recent experimental studies have shown both oscillations of exciton populations and long lasting coherence in multichromophoric systems such as photosynthetic light harvesting systems and conjugated polymers. It has been suggested that this quantum effect is due to correlations of the fluctuations of site energies among the closely packed chromophores in the protein environment. In addition to these, there is the strong possibility of correlations between site energies and transfer matrix elements. In order to understand the role of such correlations we generalize the Haken-Strobl-Reineker (HSR) model to include the energetic correlations and the site diagonal-off-diagonal correlations in a systematic way. The extended HSR model in the exciton basis is also constructed and allows us to study the dynamics of the exciton populations and coherences. With the extended model, we can provide insight into how these correlations affect the evolution of the populations and coherences of excitons by comparing to the original HSR model with uncorrelated fluctuating environments.