Carfax is a street name in Oxford and other towns. It is an old word; the OED has a citation from the mid-14th century. Clearly, therefore, it has nothing to do with cars or faxes! The OED 1357 citation spells it Carfuks. The 'car' bit means 'four' and the 'fax' bit means 'fork'; the word is related to the French word carrefour, meaning crossroads. The original meaning of carfax was 'a place where four roads meet'.
So, in the early days, a city was divided into four quarters with the carfax at the centre. Quarter meaning a district of a city was first used in around the 14th century. We've kept the word till today meaning district, but of course these days there are many more than four quarters in a city.