The charity, which provides first aid cover at all of QPR's home games, acted after the club failed to pay for more than a year. The petition gives the club until 21 February to settle the bill.
A spokeswoman for St John Ambulance said that, since the claim was served, QPR had agreed to pay the money it owes in three instalments.
She said: "We have already received two payments and the final instalment is due tomorrow. If we receive it, then the court date will be cancelled."
St John provides first aid cover for 89 clubs in England and Wales, including the majority of Premiership teams, and for matches played at the Millennium Stadium in Cardiff.
The money owed by QPR is about one month's average salary for a Championship player.
A spokesman for QPR declined to comment.