I'm not too sure it would, aksherly.
Geiger counters are very, very inefficient, (maximum is ~ 10%) and uranium only gives off low level radiation anyway.
I did my school VIth form physics project on this, leaving a Geiger counter on for a couple of weeks or so to detect background radiation, then trying to ascertain the levels of radiation given off by various lumunius toys (eg a frisbee, a 4" diameter ball, and a large patch of "Twinkle Stars"), for equal lengths of time, completely unsuccessfully - nothing significant detected above background level.
Admittedly, that was in the '70s.
I really don't think Geiger counters of the type that would be available to the public or even in schools or colleges are any more efficient nowadays.
I would imagine the detectors used in nuclear power stations are not "normal" Geiger counters.
However, as I do know that uranium beads worn by visitors to nuclear power stations have set their alarms off, perhaps you could take your glass on a little visit?