You can get zero VOC and low odour paint, or low VOC, low odour paint. Ultimately, you want - at bare minimum - low VOC paint. You also want a latex blend of paint and not oil based because of difference in fume toxicity, but in general the toxic fumes are only released as the paint dries.
I used a low VOC Behr paint with built in primer (in hind sight I'd apply the primer separately), and after one day of airing out with the birds upstairs in a well ventilated room they were put back in. It was fine. The odour might remain, but as long as all the paint is dry and the room has been aired out for at least 24 hours, it should be safe. Of course, use your own discretion and if your birds already have lung problems you'd want to take longer before letting them back in.
