  Recently, several pulsar timing array (PTA) projects have detected evidence
of the existence of a stochastic gravitational wave background (SGWB) in the
nanohertz frequency band, providing confidence in detecting individual
supermassive black hole binaries (SMBHBs) in the future. Nanohertz GWs emitted
by inspiraling SMBHBs encode the luminosity distances of SMBHBs. They can serve
as dark sirens to explore the cosmic expansion history via a statistical method
to obtain the redshift information of GW sources' host galaxies using galaxy
catalogs. The theoretical analysis of the dark siren method relies on the
modeling of the population of SMBHBs. Using a population model consistent with
the latest SGWB observations is essential, as the SGWB provides significant
information about the distribution of SMBHBs. In this work, we employ a
quasar-based model, which can self-consistently account for the SGWB amplitude,
to estimate the population of SMBHBs. We constrain the Hubble constant using
the mock GW data from different detection cases of PTAs in the future. Our
results show that a PTA consisting of 100 pulsars with a white noise level of
20 ns could measure the Hubble constant with a precision close to $1\%$ over a
10-year observation period, and a PTA with 200 pulsars may achieve this goal
over a 5-year observation period. The results indicate that modeling the SMBHB
population significantly influences the analysis of dark sirens, and SMBHB dark
sirens have the potential to be developed as a valuable cosmological probe.
