  The possibility of testing gravity theories with the help of gravitational
wave detections has become an interesting arena of recent research. In this
paper, we follow this direction by investigating the quasinormal modes (QNMs)
of the axial perturbations for charged black holes in the Palatini-type
theories of gravity, specifically ($i$) the Palatini $f(R)$ gravity coupled
with Born-Infeld nonlinear electrodynamics and ($ii$) the
Eddington-inspired-Born-Infeld gravity (EiBI) coupled with Maxwell
electromagnetic fields. The coupled master equations describing perturbations
of charged black holes in these theories are obtained with the tetrad
formalism. By using the Wentzel-Kramers- Brillouin (WKB) method up to 6th
order, we calculate the QNM frequencies of the EiBI charged black holes, the
Einstein-Born-Infeld black holes, and the Born-Infeld charged black holes
within the Palatini $R+\alpha R^2$ gravity. The QNM spectra of these black
holes would deviate from those of the Reissner-Nordstr\"om black hole. In
addition, we study the QNMs in the eikonal limit and find that for the axial
perturbations of the EiBI charged black holes, the link between the eikonal
QNMs and the unstable null circular orbit around the black hole is violated.
