My grandson has a Fiesta, even smaller than your Audi Q3!
He folds down the rear seat and then removes both wheels and lifts the bike straight into the car. He is helped by a well in the boot that means he does not have to remove the pedals.
Putting the bike in the car means that is it is safe, unlke all the other options. But it severely restricts what else you can fit in there. For you and your partner to go on hols along with your bikes will require some creative packing.
Roof racks can work, but weight is a restriction, plus the security problem. Cheap though. You may not be strong enough or tall enough to do this by yourself.
Tow bar (one with a ball, not a "receiver" like others I have seen) and rack is the route I went down. Can be expensive, depending upon options. Easy to add and remove the bikes. Weight limits are higher. No impact upon internal storage. You do not need to get an Audi tow bar fitted. They are also very expensive. A few years ago, I was quoted an unbelieveable £2400 for an A5, (even the salesman blushed). I had it done for £700 by a reputable specialist towbar company. If you are a motor electrician or have a knowledgeable friend, much cheaper. Bike racks can be almost whatever you want to pay depending upon your needs, from £200 - £650. A few years ago, I paid £420 for a tilting rack that could take two emtbs. In addition to be able to take the extra weight, you need the tyre straps to be long enough and the rails to be wide enough to take the big tyres. Let Mr Google be your friend.