The subject of this particular image is part of the Eagle Nebula in our own galaxy. So it's near enough to be gravitationally bound and not subject to the redshift of cosmic inflation.
I think my point is that because the nebula is in our own galaxy, its motion relative to Earth would be dominated by the orbital motions of the two bodies (insofar as one can consider a whole nebula a "body"). So any Doppler shift of the light traveling from the nebula to Earth would be redshifted as the galactocentric orbits of the two bodies carried them away from each other, but blueshifted as the orbits carried them towards each other.
Which is a roundabout way of also saying: the nebula and Earth are in a gravitaionally bound system (the Milky Way) where gravity overpowers whatever it is that's causing cosmic inflation. So even if their orbital motions result in a redshift as currently viewed from Earth, the amount of redshift contributed by inflation (if any) is undetectable.
That said, if you know the orbits of two bodies, and can predict their motion relative to one another, you could calculate the expected red or blue shift, as viewed from one of those bodies, at any point in their orbits. :)
