Could this mean "in the western part of the city"?
You could I suppose. Though my preference would always be to use fewer words (providing the meaning doesn’t change, of course).
But "in the west of the town" is not how it's expressed in English. We would say "in the west (or western) part of the town". (Or perhaps, "on the town's west side")
quarter should also be accepted
Why is the translation now town when before it has been city?