getprop net.dns1 or 2 in a shell. I don't think they're in any file like resource on the filesystem directly(could be wrong).
This is possibly a duplicate of this though. Check it out if you want to have it from java code. How do you get the current DNS servers for Android?
Also be advised, some Google portions will just use their own dns. Facebook app can sometimes do it too. If you're ever wondered how facebook has worked on your hotel wifi before logging in, this is how(if the wifi login is dns interception based).
EDIT: if you're looking to set the dns, you cannot do that without setting up a fake local VPN or using manufacturer specific apis(like Samsungs).