  Rare-earth Zintl compound EuIn$_2$As$_2$ has been theoretically recognized as
a candidate for realizing an intrinsic antiferromagnetic (AFM) bulk axion
insulator and a higher-order topological state, which provides a fertile
platform to explore novel topological transport phenomena. However, the axion
state has yet to be realized because EuIn$_2$As$_2$ is highly hole-doped. Here,
we synthesized a series of high-quality Ca-doped EuIn2As2
(Ca$_x$Eu$_{1-x}$In$_2$As$_2$, x = 0 ~ 0.25) single crystals to tune the Fermi
energy above the hole pocket. Our Hall measurements reveal that the isovalent
Ca substitution decreases the hole carrier density by shrinking the lattice
spacing, which is also confirmed by our first-principles calculations. We
further find that both the temperature dependence of the magnetic
susceptibility with a local maximum at the N\'eel temperature and the
topological Hall effect originating from the finite real-space spin chirality
persist in the Ca-doped samples as observed in the pristine EuIn$_2$As$_2$,
despite that the nonmagnetic Ca substitution decreases the effective moment and
the N\'eel temperature. These results show that the Ca substitution tunes the
Fermi energy while keeping the AFM magnetic structure, suggesting that the
axion insulating state may be realized by further Ca substitution.
