The Large Area Telescope onboard the Fermi Gamma-Ray Space Telescope routinely observes the Sun during its data taking. In this work we selected a sample of gamma-ray events from the Sun and we have searched for possible features in their energy spectrum to search for dark matter signatures. The capture of dark matter particles in the Sun and their subsequent annihilation via long-lived dark mediators can produce gamma rays escaping from the Sun and reaching the Earth. This process is expected to yield a feature in the energy spectrum of gamma rays from the Sun, which depends on the scattering cross section of dark matter particles with solar nucleons. We do not find any statistically significant feature in the energy spectra and thus constrain both the spin-dependent and spin-independent scattering cross sections of dark matter particles with nucleons in a dark matter mass range from a few $GeV$ up to a few tens of $GeV$ at the level of $10^{-46}$ - $10^{-45}$ $cm^{2}$ and $10^{-48}$ - $10^{-47}$ $cm^{2}$, respectively.