How is Hubble's Law related to the Doppler Effect?
It is not related, the term Doppler effect was borrowed from an unrelated effect.
the Doppler effect concerns the wave movement of the medium (air) in response to pressure (sound). Hubble's law proposes that the medium (space) is distorted by energy/mass.
To accept this requires you to believe that space or nothing is the medium and that the nothingness is being distorted.