X2. On my signature, there is a link on what can happen when a chicken infects a quail. They need to stay apart. I wish there was a better answer, but when you go to the link, you will see that the member did not have his quail and chickens "together", but the chickens still infected the quail through what he believes was the use of gloves from one area to another. Chickens have a much stronger immunity system than quail, and can live a long time as a carrier of a disease and never show any symptoms.
Now, there may be some that will chime in and say, "I keep my quail and chicken together and I don't have a problem". In those cases, they may practice better bio security procedures than most.
Even if you have no problems at first, disease can be carried by bugs they eat, or incubate in feces or wet soil. At any point that you own both chickens and quail, you should practice bio security.
Button quail mate for life and should be kept in pairs, the only way you can keep them with coturnix is in an aviary style setting and make sure everyone has a hiding place. Quail are very territorial and can be quite aggressive. A lady that buys chicks from me sometimes has several nice aviaries and she keeps a trio of coturnix and a couple of pairs of buttons in each one.
This is a full grown button for size comparison
This is a grown coturnix