When you say 'in chicken coops,' do you mean with chickens for company? This won't work as the chickens will attack or even kill them. A small chicken coop and run might be suitable but quails have different needs to chickens, and the run would have to have 1/2" mesh, not the larger kind often found for chickens, or they would escape. They would also need dry conditions to live outside, ie as a minimum a roofed area with protection at the sides from wind and rain. They don't need nestboxes but need a place to go in the coop where they can shelter together and some places to hide in the run. They can tolerate cold but damp is dangerous especially in winter. I used to put mine outside from April to October, depending in the weather, and kept them in a dry cage inside during the winter,
Given suitable conditions and good weather, they can go to new homes from about 4 weeks old, ie when off heat in the brooder. Again this depends on several things, mainly how well they are readily to cope with the changed conditions outside compared with in the brooder. They will need to have been off heat entirely in the brooder for several days at least before they go outside, as temperatures do drop at night compared with indoors.
You would be best to keep at least two of them to introduce to the butterfly house, as introducing one single bird of any poultry species is very difficult. Newcomers always get bullied by the adults already there, males in particular regard newcomers as a challenges especially other young males,