Oh, yeah ... what you're proposing is definitely illegal. There's a season for geese in Tennessee (Sept 1-15), along with all kinds of laws and restrictions on hunting just about everything from deer to bullfrogs (do a "Google" search under "Tennessee hunting laws"). If you're going to shoot geese, you need a hunting license and you need to do it in-season.
Geese are a big problem in the Northeast, particularly in non-hunter-friendly urban areas. Golf courses and public parks are near overrun with them. And you pretty much can't shoot them. Some places use "cherry bombs" or other noise makers; generally most effective are border collies, who harrass them enough that they go elsewhere.
It's said that the geese that hang around in Winter in the Northeast are "feral" descendents of geese that hunters used to keep as live decoys. They don't have any migratory instinct. (you could try that as a legal defense ... they weren't "migratory" wildlife ... but I wouldn't count on it).
Some states allow farmers to go after otherwise-season-only game if they're harming their crops .... like deer in orchards.
That said, geese are pretty big and tough; with an airgun, only a solid head-shot will do (not impossible with a good airgun at close range). Otherwise, you'd do about as well with a hatchet. But, please note that ... considering the legal implications ... I'm not actually recommending anything here.