Hi there,
What a worrisome thing you are dealing with my dear, but the best way forward is to keep your head and be relaxed with your daughter. The very first thing I thought of is this 'green bird' myth. Which prevades all sorts of cultures and I even heard the American Gnostic Templars refer to it in a huge way (for some reason their site won't open for me at the moment)
http://www.yeziditruth.org/the_peacock_angel
This will give you some information, of course with all things there is a duality, the green bird can also be referred to as the green dragon and take on a different feeling if you catch my drift. If your daughter is able to match the bird to a colour that would be helpful.
Its could always be the famous blue bird of happiness, don't think it hurts though. As the others all wisely said, tell her for certain that she can say NO I DONT LIKE THAT.
please let us know if she tells you anything more.