  The consideration of a 1 + 3 covariant approach to cold dark matter universe
with no shear cosmological dust model with irrotational flows is developed in
the context of f (G) gravity theory in the present study. This approach reveals
the existence of integrability conditions which do not appear in non-covariant
treatments. We constructed the integrability conditions in modified
Gauss-Bonnet f (G) gravity basing on the constraints and propagation equations.
These integrability conditions reveal the linearized silent nature of
quasi-Newtonian models in f (G) gravity. Finally, the linear equations for the
overdensity and velocity perturbations of the quasi-Newtonian space-time were
constructed in the context of modified f (G) gravity. The application of
harmonic decomposition and redshift transformation techniques to explore the
behaviour of the overdensity and velocity perturbations using f (G) model were
made. On the other hand we applied the quasi-static approximation to study the
approximated solutions on small scales which helps to get both analytical and
numerical results of the perturbation equations. The analysis of the energy
overdensity and velocity perturbations for both short and long wavelength modes
in a dust-Gauss-Bonnet fluids were done and we see that both energy overdensity
and velocity perturbations decay with redshift for both modes. In the limits to
{\Lambda}CDM , it means f (G) = G the considered f (G) model results coincide
with {\Lambda}CDM .
