Gravitational-wave (GW) cosmology has been traditionally done via the standard-siren approach, relying on the ability to obtain distance information from the GW data, and the redshift information using electromagnetic (EM) survey observations or archival galaxy catalogs. Here, we implement a new way to measure the Hubble constant without an EM counterpart through the use of the binary-Love relations. These relations govern the tidal deformabilities of neutron stars in an equation of state insensitive way. We leverage the fact that the presence of matter in the merger signal in conjunction with the use of binary love relations allows a measurement of the redshift from the GW data. We implement this approach in both real and synthetic data through a Bayesian parameter estimation study in a range of observing scenarios. We find that for the LIGO/Virgo/KAGRA design sensitivity era, this method results in a similar measurement accuracy of the Hubble constant to those of current-day dark-siren measurements. For third-generation detectors, this accuracy improves to ≲ 10% when combining measurements from binary neutron star events in the LIGO Voyager era, and to ≲ 2% in the Cosmic Explorer era.