  We explore the topological susceptibility at finite quark chemical potential
and zero temperature in two-color QCD (QC$_2$D) with two flavors. Through the
Ward-Takahashi identities of QC$_2$D, we find that the topological
susceptibility in the vacuum solely depends on three observables: the pion
decay constant, the pion mass, and the $\eta$ mass in the low-energy regime of
QC$_2$D. Based on the identities, we numerically evaluate the topological
susceptibility at finite quark chemical potential using the linear sigma model
with the approximate Pauli-Gursey $SU(4)$ symmetry. Our findings indicate that,
in the absence of $U(1)_A$ anomaly effects represented by the
Kobayashi-Maskawa-'t Hooft-type determinant interaction, the topological
susceptibility vanishes in both the hadronic and baryon superfluid phases. On
the other hand, when the $U(1)_A$ anomaly effects are present, the constant and
nonzero topological susceptibility is induced in the hadronic phase, reflecting
the mass difference between the pion and $\eta$ meson. Meanwhile, in the
superfluid phase it begins to decrease smoothly. The asymptotic behavior of the
decrement is fitted by the continuous reduction of the chiral condensate in
dense QC$_2$D, which is similar to the behavior observed in hot three-color QCD
matter. In addition, effects from the finite diquark source on the topological
susceptibility are discussed. We expect that the present study provides a clue
to shed light on the role of the $U(1)_A$ anomaly in cold and dense QCD matter.
