Do you use gesso for oils or acrylic painting?
There are various recipes for gesso, with the majority involving some kind of chalk or plaster mixed with an adhesive binder. The exact components vary by brand and application.
Due to the adhesive binder (glue), most gesso will have a noticeable odor. Acrylic gesso often smells strongly of ammonia when it is fresh and wet. And since you typically use gesso over a large surface area, it is recommended to work with gesso in a ventilated work space for your own comfort and good health.
Personally, I would use the same precautions when working with gesso as you would use when working with acrylic paints. Pick a well-ventilated work space away from your birds and keep them away until the paint is dry and any odors have cleared out.