In special relativity, redshifts arise directly from velocities. It was this idea that led Hubble in 1929 to convert the redshifts of the “nebulae” he observed into velocities, and predict the expansion of the universe with the linear velocity-distance law that now bears his name.
The general relativistic interpretation of the expansion interprets cosmological redshifts as an indication of velocity since the proper distance between comoving objects increases. However, the velocity is due to the rate of expansion of space, not movement through space, and therefore cannot be calculated with the special relativistic Doppler shift formula.