If my business files chapter 11 bankruptcy, must we pay quarterly fees to the court?

Yes, there are quarterly fees that must be paid, not exactly to the court, but to the United States Trustee program for every quarter that your chapter 11 case is open. The fees are not too bad, they are based on how much money you disburse to your vendors and others during the bankruptcy case. 

The U.S. Trustee's office is the "watchdog" of the bankruptcy system and the fees that they collect are intended to make the UST program self-supporting or at least defray the cost of the program.