Does time also pass more slowly for a galaxy that is traveling at relativistic speeds, where the speed difference is due to Hubble expansion? If time passes more slowly, relative to Earth, for a traveler at relativistic speeds, say at .8c, traveling between stars inside this galaxy, does time also pass more slowly for a distant galaxy that is moving away at relativistic speeds, where the speed difference is due to the Hubble expansion of space?
Yes, but it has little to do with the Special Relativistic time dilation of a moving body - in the sense that recession velocity is due to the expansion of space between galaxies rather than their relative motions.
The cosmological expansion leads to a redshift $z$ and also to time dilation by a factor $(1+z)$. i.e. Those distant clocks appear to run slower to us by a factor $(1+z)$. And vice versa - our clocks would appear to run slow from the perspective of an observer in a distant galaxy.
The time dilation effect has been observed in the light curves of high redshift supernovae (Blondin et al. 2008) and the durations of gamma ray bursts (Zhang et al. 2013) and is consistent with this prediction. This is an important observation since it is very difficult to account for this time dilation effect using a &quot;tired light&quot;  explanation.