There is a possibility of that if space itself was not expanding and something could be relatively expanding at the same rate as our galaxy. Then you would never see a red shift from our galaxy.

But when we say universe is expanding, it's also the space between galaxies that is expanding. So, everything is moving away from each other. I highly doubt if we will ever find anything not moving relative to us because even the space between earth and anything else in the galaxy is expanding constantly.