  Dark matter and inflation are two key elements to understand the origin of
cosmic structures in modern cosmology, and yet their exact physical models
remain largely uncertain. The Weyl scaling invariant theory of gravity may
provide a feasible scheme to solve these two puzzles jointly, which contains a
massive gauge boson playing the role of dark matter candidate, and allows the
quadratic scalar curvature term, namely $R^2$, to realize a viable inflationary
mechanism in agreement with current observations. We ponder on the production
of dark matters in the Weyl $R^2$ model, including the contribution from the
non-perturbative production due to the quantum fluctuations from inflationary
vacuum and perturbative ones from scattering. We demonstrate that there are
generally three parameter ranges for viable dark matter production: (1) If the
reheating temperature is larger than $10^3~\mathrm{GeV}$, the Weyl gauge boson
as dark matter can be produced abundantly with mass larger than the inflation
scale $\sim 10^{13}~\mathrm{GeV}$. (2) Small mass region with
$3\times10^{-13}~\mathrm{GeV}$ for a higher reheating temperature. (3)
Annihilation channel becomes important in the case of higher reheating
temperature, which enables the Weyl gauge boson with mass up to
$4\times10^{16}~\mathrm{GeV}$ to be produced through freeze-in.
