Cosmological redshift is based on the idea that the universe is expanding. When the universe doubles in size, or scale factor, the wavelength of light doubles. But the Milky Way is not expanding so my guess is that there is no cosmological redshift within the Milky Way? There is of course Doppler redshift.
Having seen the first 5 answers, which seem to confirm my guess, I am now going to add a corollary. The Doppler effect has almost nothing to do with cosmological red shift, outside our local cluster of galaxies. Is that also correct? Which makes Hubble very lucky.
Best Answer
Your guess is correct. The constituents of the Milky Way, and also close dwarf galaxies like the Magellanic clouds form a gravitational bound system and have therefore, seen as a single system, decoupled from the Hubble flow. This happened because the average energy density within this system at one time in the cosmological history became much larger than the average energy density of the universe (One can check out reviews about structure formation for more details). (Of course the whole system is still part of the expansion in the sense that the distance to galaxies far away increases.) Therefore there is no cosmological redshift observed within this system, for example from Earth, for any stars within or other constituents of these galaxies.