Here is the code for Unincorporated Jefferson County.
I will locate the one for Birmingham later.
Article 11 Section 1109
No livestock, poultry or fowl shall be permitted in any residential zoning district.
Here is the PDF link...
Found out in Unincorporated Jefferson County Alabama, If you do not have a zone of A-1 or other agriculture zones, you can only own dogs and cats as domesticated animals. We are having to get a variance for my 3/4 acre land to keep my geese, goats, and soon to be chickens.