  More recently, Fernandes \cite{Fernandes:2023vux} discovered analytic
stationary and axially-symmetric black hole solutions within semiclassical
gravity, driven by the trace anomaly. The study unveils some distinctive
features of these solutions. In this paper, we compute the gravitational waves
emitted from the \ac{EMRI} around these quantum-corrected rotating black holes
using the kludge approximate method. Firstly, we derive the orbital energy,
angular momentum and fundamental frequencies for orbits on the equatorial
plane. We find that, for the gravitational radiation described by quadrupole
formulas, the contribution from the trace anomaly only appears at higher-order
terms in the energy flux when compared with the standard Kerr case. Therefore,
we can compute the EMRI waveforms from the quantum-corrected rotating black
hole using the Kerr fluxes. We assess the differences between the EMRI
waveforms from rotating black holes with and without the trace anomaly by
calculating the dephasing and mismatch. Our results demonstrate that
space-borne gravitational wave detectors can distinguish the EMRI waveform from
the quantum-corrected black holes with a fractional coupling constant of $\sim
10^{-3}$ within one year observation. Finally, we compute the constraint on the
coupling constant using the Fisher information matrix method and find that the
potential constraint on the coupling constant by LISA can be within the error
$\sim 10^{-4}$ in suitable scenarios.
