Well, one of the best ways to get him to allow you a few chickens (I advise to start with a small, easily manageable number. You can let chicken math take over later after he's totally warmed up to them) is to get fresh-laid eggs from a farm stand or a friend with hens. Cook him up some fresh-laid scrambled eggs, and, assuming he likes eggs, and he's never had this experience, he'll be amazed at the difference.
So you'll be selling him on a promise of good eating, first.
Next, you might demonstrate that you know how to design & build a coop for these few chickens and can contain them in a simply made run. Tell him how you would go about frequently cleaning the coop & run, and how you could composed the manure to use around vegetable or flower beds later. Educate yourself well about chicken-keepuing, and then educate him. I used this technique on my husband, and it worked well.
I hope this helps.