I doubt anybody will notify you specifically but you have several ways to look for them yourself. There is a Create/Join a Tournament
which you can check as often as you wish for new tournaments. Then you can also use the Tournament Tab
and more exactly Tournament Finder where you just have to pick "Quadruples" as Type and "Open" as Status.
When I start thinking of it then second option is better for you because you are searching for a certain type of tournament. Checking it daily shouldn't be a problem and I'm pretty sure that within a week you'll find a tournament to join (unless you are very picky).