In the framework of the generalized Brans-Dicke theory of gravity, time dependence of various cosmological parameters has been determined in the present study for a spatially flat, homogeneous and isotropic universe filled with pressure-less matter. Mathematical formulations have been carried out with the help of two models, based upon two different expressions for the scale factor. In the first model, the entire matter content (dark matter + baryonic matter) of the universe has been assumed to be conserved. A smooth transition from a state of decelerated expansion to a state of accelerated expansion of the universe has been obtained from an exact solution of the field equations, without incorporating any parameter in the theoretical formulation that represents the dark energy. Time dependence of the scalar field has been determined from this solution with the help of astrophysical characteristics of the expanding universe. The nature of dependence of the Brans-Dicke parameter upon time and also upon the scalar field has been found. The Brans-Dicke parameter has been found to have a small negative value and it becomes more negative as the scalar field decreases with time. It has been found in the present study that the gravitational constant, which is reciprocal of the scalar field parameter, increases with time. In the second of the two models discussed here, an ansatz has been assumed regarding the mode of change of the dark energy content of the universe with time. Using this model, the time dependence of the densities of matter and dark energy and also the density parameters corresponding to these two constituents of the universe has been determined.