Primordial black holes generated in the early universe are considered as a candidate for dark matter. The scenario where neutron stars capture primordial black holes gives rise to a series of observable consequences, allowing neutron stars to play the role of dark matter detectors. Gravitational waves from primordial black holes inspiralling inside neutron stars are detectable for ground-based detectors, providing a unique insight into the neutron star structure. We generate gravitational wave templates for such systems with different equations of state of neutron stars by solving the Einstein equations inside neutron stars, and taking the dynamical friction, accretion, and gravitational radiation into account. We find that Einstein Telescope can distinguish the equation of state of neutron stars by detecting gravitational waves from primordial black holes inspiralling inside neutron stars within the Milky Way galaxy.