For us to be able to reside at such a tremendous distance from where the big bang occurred, we'd need to travel at an average of 11.7 / 13.7 (or 85%) of the speed of light from the moment of the big bang, directly away from the direction of the observed quasar, for the entire 13.7 billion years of our universe's existence.
I don't think either of those scenarios is possible (the quasar traveling at light speed, or our galaxy's matter averaging 85% light speed or more, for all the universe's 13.7 years' existence.) This causes me to doubt (a) the big bang theory, (b) the age of the universe, or (c) my own reasoning. Can you help get me out of this paradox?
"there" for the last 13.7 billion years. Indeed we believe that if we were able to look outwards from a distant galaxy, the Universe would look the same as it does from Earth. And from that galaxy the visible Universe would extend 13.7 billion light years outwards, as it does for us.