Transition frequencies of Fe II ion are known to be very sensitive to variation of the fine-structure constant $\alpha$. The resonance absorption lines of Fe II from objects at cosmological distances are used in a search for the possible variation of $\alpha$ in the course of cosmic time. In this paper we calculated the dependence of the transition frequencies on ${\alpha }^2$ ($q$ factors) for Fe II ion. We found corrections to these coefficients from valence-valence and core-valence correlations and from the Breit interaction. Both the core-valence correlation and Breit corrections to the $q$ factors appeared to be larger than had been anticipated previously. Nevertheless our calculation confirms that the Fe II absorption lines seen in quasar spectra have large $q$ factors of both signs and thus the ion Fe II alone can be used in the search for the $\alpha$ variation at different cosmological epochs.