Is there cosmological redshift within the Milky Way? 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.
 A: Even if the Milky Way is expanding with the Hubble flow (and most cosmologists believe that it isn't), the expansion would be difficult to measure.
The size of the Milky Way $d$ is approximately $10^{21} \;\text{m}$.
Using $$v=Hd$$
with Hubble's constant in SI units of about $2\times 10^{-18} \;\text{s}^{-1}$
means that even a star far away, in the Milky Way, would have a redshift corresponding to a motion of about $2000 \;\text{m}\,\text{s}^{-1}$.  However peculiar velocities of stars are typically a hundred times higher than this, for example the sun is thought to be moving relative to the Milky Way at $250 \;\text{km}\,\text{s}^{-1}$.
Cosmological redshifts are used to determine distances of stars etc... that are far enough away that the peculiar velocities can be ignored.
It would be interesting to see the results of a future experiment that attempted to 'average out' the peculiar motions of millions of stars within the Milky Way, to see if there is a cosmological redshift.
It seems that at the moment the consensus amongst cosmologists is that it doesn't exist.
A: There is not.
Cosmological expansion (and therefore redshift) are caused by general relativity. We understand this through the Friedmann–Lemaître–Robertson–Walker metric, which describes a homogeneous, isotropic universe.
But you may notice that the Universe doesn't appear homogeneous or isotropic. Some parts are denser than others, like galaxies. And when you look around, you might even argue that the galaxies aren't exactly the same, so it cannot be isotropic.
But if you look at the Universe from the point of view of a huge Gigaparsec creature (1000x larger than Megaparsec galactic systems), the galaxies and inhomogeneities appear like sand on a beach (we are 1000x larger than millimeter grains), or atoms in air. This is a way to illustrate that at a large enough scale, the Universe does appear homogeneous and isotropic, and can be well-described by the FLRW metric.
The cosmological expansion of the Universe depends on assumptions, and those assumptions are valid at large scales, but not small scales. Hence, the prediction of general relativity is that there will be expansion on those large scales where those assumptions hold true, but not necessarily expansion on the small scales where these assumptions do not hold true. And if you had a large enough computer to simulate the equations of general relativity in the Milky Way, you would not find cosmological redshift (most cosmologists believe).
A: 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.
A: If everything were expanding with the Big Bang expansion, stars from each other , planetary systems, atomic systems, nuclear systems there would be no possibility of measuring the expansion, because the units scientists use to measure would also be changing  (without our being able to perceive it) following the expansion.
The hypothesis that allows to measure an expansion is that the bound states of the four known forces,  cannot be affected by the expansion. This allows to treat galaxies, and the stars within them, as gravitational bound states ,not changing size, and allows for expansion and the infrared shifts one measures.
So it is not a matter of consensus and beliefs, it is an inevitable  axiom in the  model, the Big Bang, that fits the data observed. At the moment it is validated by all the known observations. Thus the hypothesis is extended to the bound states of the other three forces, which bind more strongly than gravity.
Maybe future observations and measurements might show discrepancies with this hypothesis, then a modified Big Bang would have to be used , though I think the argument of "how one could observe any change when the units change" will have to be addressed.
A: For a technical discussion of this type of issue, with numerous references to other work, see
"On the influence of the global cosmological expansion on the local dynamics in the Solar System"
https://arxiv.org/abs/gr-qc/0602098
and
"The Influence of
the Cosmological Expansion on Local Systems" https://arxiv.org/abs/astro-ph/9803097.
These papers discuss the problems of defining measurements within general relativity, and show that the overall cosmological expansion has extremely small effects (but not zero) on the scale of the solar system.
