Greetings from Ireland
The good news is don't worry - you will have no problem getting tickets. Round 4 qualifiers are not a huge crowd-puller, in fact the only games sold-out are the 2 actual finals and any games involving Dublin, although I attended all Dublin games last year and most were not sold-out. For the Leinster final against Westmeath, there must've been 20,000 empty seats despite the day having glorious sunshine.
The system of distributing tickets among counties only happens for the big finals, and they aren't util September. I'm a member of the GAA and a lifelong fan.
So just wait until July, you'll be able to buy your tickets online or at the stadium on the day. I guarantee you a Round 4 qualifier will not be sold-out. If the game is in Dublin's Croke Park (which is GAA HQ) then it will likely be half-empty (which is still an impressive 40,000 in attendance!).
The qualifiers are for teams who have already been beaten or knocked out of their provincial championships and this is the 'back-door' last chance for them to get into the quarter finals, so all is at stake, but there'll be plenty of tickets available.
You wont need to contact the club in Atlanta.
Enjoy our national games, try and check out the GAA museum beforehand.
le meas/with respect
Stephen McPhilemy
Rick Steves Ireland tour-guide
Dingle, Derry & Dublin