Modified gravity and massive neutrino cosmologies are two of the most interesting scenarios that have been recently explored to account for possible observational deviations from the concordance $Λ$-cold dark matter ($Λ$CDM) model. In this context, we investigated the large-scale structure of the Universe by exploiting the \dustp simulations that implement, simultaneously, the effects of $f(R)$ gravity and massive neutrinos. To study the possibility of breaking the degeneracy between these two effects, we analysed the redshift-space distortions in the clustering of dark matter haloes at different redshifts. Specifically, we focused on the monopole and quadrupole of the two-point correlation function, both in real and redshift space. The deviations with respect to $Λ$CDM model have been quantified in terms of the linear growth rate parameter. We found that redshift-space distortions provide a powerful probe to discriminate between $Λ$CDM and modified gravity models, especially at high redshifts ($z \gtrsim 1$), even in the presence of massive neutrinos.