No. Nothing sticky. Too easy to have an accident and a bird get stuck to it. You can slightly open the screen when the birds are safely in their cages... turn the lights off, make it dark. The flies should go to the light and get back outside.

Or:

Vinegar and Dish Soap Trap:
.
A simple and effective method for fruit flies. Fill a jar or glass with apple cider vinegar and add a few drops of dish soap to break the surface tension. The smooth sides of the glass prevent flies from escaping.