The Reds are fuming after referee Steve Walsh denied them a 49th-minute try and failed to take action on Moore's left-arm upper-cut on Quirk at a ruck on the goal line on Friday night.
Will Genia had dived over in the corner but the try was scratched as Walsh ruled Quirk illegally held Moore before the Wallabies rake lashed out.
Queensland coach Richard Graham was livid post-match and felt Moore should have been shown a yellow or red card for his punch.
While the citing commissioner was running his rule over the match, Graham indicated the Reds were keen to have the matter investigated.
Brumbies coach Laurie Fisher commended Walsh for his hard-line stance in clamping down on holding at the ruck, and felt Quirk had cost his side a try.
But Graham and Queensland skipper James Horwill whole-heartedly disagreed.
"I would see Stephen Moore as part of the breakdown and I think Ed Quirk is entitled to hold him," Graham said.
Horwill said: "If it is a punch, the rules are the rules and you have to go to the rule book (and send him off)".
Walsh's decision enraged the 30,000-strong Brisbane crowd and a large group of fans heckled Moore on the sideline which prevented a post-match Foxsports panel interview.
The Brumbies withstood a second-half onslaught by Queensland's transformed pack to put the Reds' Super Rugby finals hopes on the brink.
It was the Reds' third loss this season by three points and has them 11th - nine points behind the ACT side - and slipping out of play-off contention heading into next weekend's bye.
In a topsy-turvy clash where the Brumbies dominated the first half before the Reds pack - boosted by the injection of 100-gamer Saia Fainga'a - turned the tables in the second.
The set-piece turnaround was highlighted by a remarkable pushover try to Quirk from a tight-head scrum win in the 63rd minute that sparked the Reds revival and set up a tense finish.
The Brumbies were far slicker, faster and smarter in the first half and went into the sheds at 20-10 but were kicking themselves the lead wasn't 20-plus after regularly letting the listless Reds off the hook.
Queensland shot themselves in the foot from the outset with Quade Cooper gifting Scott Fardy the opening try after Robbie Coleman jolted the ball free.
The visitors were in control at 20-3 after half an hour when Nick White and Jesse Mogg - the two players the Reds targeted pre-match - created the second try to Joe Tomane.
