Robins boss Derek McInnes is currently without the services of long-term absentees George Elokobi and Greg Cunningham, while versatile defender Liam Fontaine also picked up an injury in the midweek defeat to Burnley.
England Under-21 international Briggs, who has been second choice behind John Arne Risse at Craven Cottage this season, has joined on a one-month deal to help fill the void.
McInnes said: "It's common knowledge that injuries have left us short in defence over the past few weeks and we're delighted to get Matthew on board.
"He'll certainly offer us cover at left-back while Greg [Cunningham] gets himself back to full fitness. Greg is making good progress, but he's nowhere near yet.
"Matthew is quick, energetic and hungry. He's a very highly-rated defender who can play in a number of positions and this loan move works for all parties concerned."
Briggs, who has previously spent time on loan at Championship outfit Peterborough United, debuted for Fulham in 2007 aged 16, making him the Premier League's youngest player of all time.