Chickens don't have a gag reflex. Liquid coming/pouring from the beak, aspiration. Doesn't really matter where she was located in the brooder.
As to why their was fluid coming from the beak...impacted crop, had something stuck in her throat...a bit of speculation there, unless it was...
Photos of the brooder?
Sounds like she may have aspirated.
If the chicks are on shavings, then I would provide chick grit, sometimes chicks will eat bits of bedding.
Yes, it's possible for a chick(en) at any age to have a crop problem.