  We investigate the structure of quark stars in the framework of $f(R)= R+
\alpha R^2$ gravity using an equation of state for cold quark matter obtained
from perturbative QCD, parametrized only by the renormalization scale. We show
that a considerably large range of the free parameter $\alpha$, within and even
beyond the constraints previously reported in the literature, yield
non-negligible modifications in the mass and radius of stars with large central
mass densities. Besides, their stability against baryon evaporation is analyzed
through the behavior of the associated total binding energies for which we show
that these energies are slightly affected by the modified gravity term in the
regime of high proper (baryon) masses.
