Yes, but one issue is that the amount of redshift depends on the motion of the emitter, so we would have to artificially assign some four-velocity to your surroundings in order to give them some redshift. There doesn't seem to be a "natural" choice for how to do this.
TLDR: redshift depends not only on the position of the source, but also its velocity.
TLDR: redshift depends not only on the position of the source, but also its velocity.