Can’t answer based on having Q2, but there are a lot of delays at the moment due to the WLTP emissions testing. If you ordered now, I would imagine Feb or more likely March is about the timescale. Others in here have ordered cars back in April / May and a lot still waiting for order to be accepted or build slot provided. Seems as though starting to slowly get moving but bet there is a big backlog. Look through this forum for threads around WLTP. And be careful of dealers promising quick delivery which may just be to get you to sign up
All new cars sold from 1st Sept 2018 have to have been WLTP tested (I'm assuming thos eto be new builds as opposed to existing stock). I thought about a Q2 and test drove one last week, all you could order then were the 1.4 Diesel variants, nothing else. Dealer did ring back the following day and advise that Audi have (apparently) opened the books if you want a 2.0 petrol (having told me the day before that all 2.0 engines were destined for the new SQ2 just released at the Paris show).
Until the engines have been WLTP tested, my understanding is that they are not being built, hence the backlog. My gut feel is that most diesels have bene tested following the uproar about diesels a while back and that the petrol engines are undergoing WLTP testing now, but that (hopefully) the end of that cycle is in sight. All you can do is stay in touch with the dealer if you want a Q2 is the advice I was given.