You don’t have to wait for the Gamemaster to roll an Opportunity to try to remove a Complication. If you want to get out from under the Complication’s negative effects, you (or another Crewmember) can put together a recovery roll with the appropriate Attribute, Skill, etc. against a dice pool featuring the Complication and a difficulty die. Of course, not all Complications are easily dismissed; it’s going to be pretty hard for Kaylee to deal with a Broken Arm all by herself and Jayne is more likely to talk himself into more trouble after he Broke the Nose of the Lawman’s Son. It’s probably best to leave the medical treatments to Simon and the talkin’ to Inara!
If you’re successful at your recovery roll, remove the Complication completely. If you fail, step back the Complication die to represent that you’ve made some progress patching up a broken limb or smoothing things over with the local law enforcement types. There’s a few limits on this: you may only attempt to remove a Complication once per scene and the GM may rule that it’s impossible for you to actually make a recovery roll at all. It’s pretty tough to clean up an Oil Spill d8 without the proper gear to mop up the mess!
It’s also possible that you might roll a jinx on a recovery roll, resulting in the situation worsenin’ for you and your Crew. If you roll a jinx on a successful recovery roll, the GM can hand over a Plot Point to introduce a new Complication related to your attempt to improve your situation—your Broken Arm d8 is now Splinted Limb d6. If you roll a jinx on an unsuccessful roll, your Complication gets worse by one step for every jinx you rolled without the GM havin’ to pay any Plot Points at all, as your failed attempts to make things better result in you aggravatin’ the situation.