Greys are VERY intelligent. They will learn routines and schedules in regards pooping, eating, playing and all other times. Heck, they will even tell you it's time to feed the dogs if you run a little late doing it.
I agree with Daves thoughts on this subject. Dayo will poop in his cage if we do not take him out with in the first 30 minutes or so first thing in the morning. If we do get him out first thing, then he will go once we place him over the kitchen sink (makes it easy to just rinse away). But, once out, ALL area are fair game for when ever he feels the need to let one go.
I guarantee anyone that leaves their Grey caged for most the day without taking them out will find a morning bomb in the cage sooner or later, as Dave said. Of course, they can try and aim it out of the cage though, which really annoys me when Dayo does that...Talk about a messy carpet first thing in the morning you gotta clean upu!!!