Gillbee was pulled over at Broadbeach.
The reading is considered a mid-range drink driving offence in Queensland.
In a statement on Sunday, the Suns said the forward notified the club immediately and cooperated fully with Gold Coast Police.
"We have spoken with Jacob this morning and he is very embarrassed by the situation he finds himself in, and well aware he has let himself and the club down by making this unacceptable individual choice," Gold Coast football manager Marcus Ashcroft said.
The Suns have been given a mini-break by their coaches over the Australia Day long weekend and are due back at training on Tuesday.
Gillbee has played three games for the Suns, including two in 2012.
He was selected with pick No.49 in the 2010 NAB AFL Draft.
Meanwhile, Carlton player Mitch Robinson is being investigated by his club after an apparent scuffle during the Big Day Out at Flemington on Saturday.
Photographs showing Robinson in an incident at the music festival have featured on Twitter.
One eye witness told News Limited Robinson was 'targeted' by trouble makers.
