Edit to add: It looks like the ordinance @KikisGirls found (two posts below this) specifically addresses fowl (where this one addresses livestock in general). It would seem hers is the applicable ordinance. (Good job @KikisGirls !)
This if from the City of Desoto Code of Ordinances 2.1000(a)
"(a) It shall be unlawful for any person to keep on any premises under such person's control within the city limits any livestock, except pigs and hogs, in such a manner that the same will be located closer than one hundred and fifty feet (150') distance from any person's living quarters, other than his own."
That means that you have to have your coop and run 150 feet from anyone else's house. If you live on a normal size lot in the city, that's not possible.
However, while chickens are livestock, quail are not (they are game birds). I didn't see anything in the code about game birds. So you may be able to keep quail.