  We present a formalism to calculate the non-linear matter power spectrum in
modified gravity models that explain the late-time acceleration of the Universe
without dark energy. Any successful modified gravity models should contain a
mechanism to recover General Relativity (GR) on small scales in order to avoid
the stringent constrains on deviations from GR at solar system scales. Based on
our formalism, the quasi non-linear power spectrum in the
Dvali-Gabadadze-Porratti (DGP) braneworld models and $f(R)$ gravity models are
derived by taking into account the mechanism to recover GR properly. We also
extrapolate our predictions to fully non-linear scales using the Parametrized
Post Friedmann (PPF) framework. In $f(R)$ gravity models, the predicted
non-linear power spectrum is shown to reproduce N-body results. We find that
the mechanism to recover GR suppresses the difference between the modified
gravity models and dark energy models with the same expansion history, but the
difference remains large at weakly non-linear regime in these models. Our
formalism is applicable to a wide variety of modified gravity models and it is
ready to use once consistent models for modified gravity are developed.
