Nope.
We have accounts of pigeons living in urban centres as early as the 4th millenium BC in Mesopotamia in the earliest cuneiform texts. We know that chickens (from wild red jungle fowl) were domesticated much earlier in Asia (from genetic analysis) but our first written evidence of deliberate bird domestication relates to Mesopotamian pigeons.
Pigeons were certainly residing in Roman Cities, living alongside the human inhabitants and being used by them for food, sport etc. They were there in large enough numbers to cause problems, such as defecating and nesting in inconvenient places.
However, they would not have been present in the huge numbers we see today. This is because since the 1950s we have seen an explosion in the number of humans living in cities and in the amount of food we produce and throw away. Animals which are able to exploit this ecological niche of living on our left-overs have consequently undergone population booms.
One striking example is the grey squirrel in England which was a newly introduced and rarely encountered species as recently as the 1890s and was still not very prevalent by the 1950s but which is now a very frequent inhabitant of urban parks and gardens to the extent that it needs to be controlled to protect the native red squirrel.
https://anatomie.unibas.ch/IntegrativeBiology/haag/Culture-History-Pigeon/feral-pigeon-haag.html
http://extension.arizona.edu/sites/extension.arizona.edu/files/pubs/snr07041f.pdf
Follow up question: were there any rodents/pests common in ancient cities that we no longer see today?
Many Italians, mostly Southern Italians, used the pigeon as a food source. If you look at the history of Southern Italy and how poor off they were, they ate what they could find. Even as recently as post WWII they were adding pigeon meat to tomato sauces because it was cheap and times were hard. I would venture to say a part of the reason there are so many pigeons in big cities is because they aren't being consumed (or shot by bored kids).
http://www.pigeoncontrolresourcecentre.org/html/about-pigeons.html