Hi Lee,
If you want it quiet then Gozomark is right about sugesting Gozo. Bear in mind though that after you have landed your forward journey will involve a drive to the ferry to get to Gozo and then another short journey to your hotel. This could be a bit tiring especially if you arrive late at night. I think you can avoid this by using the helicopter, if this service is still available, but I think that would cost you extra.
I would certainly recommend hiring a car especially knowing you have a 16 month old. I remember when ours was that age and it's amazing what you need to carry with you. A car will also let you explore more and since they drive on the left (usually!), it shouldn't be aproblem driving over there. You can also take a car on the ferries therefore allowing you to explore both major Islands.
Car hire is relatively cheap and when we were in Malta recently we hired a Hyundai Getz with aircon (recommended in the hotter weather), for £15 a day and that included full extra insurance cover which waived any excesses in case of any incidents (also recommended). We normally hire from Sundrive Car Hire who have a web page at,
http://www.sundrive.com.mt/
I think you will have to pay a bit more than we did to hire because rates go up as the weather improves but you can check their rates on the net. They will deliver and collect your car as necessary and I have really found them to offer an excellent service.
There are places on the mainland that are also suitable for what you want but we tend to stay in places such as Sliema which are quite busy. However, there are hotels such as the Salmen Palace that are outside the busy centres which I would imagine are also fairly quiet.
Hope this helps,
Trev