Any decent quick places to grab dinner around the Staples Center?
Not really. I've been faced with this situation a lot, and usually with limited time. Two places to avoid completely: Liberty Grill and that Italian place next to the car wash.
Other options nearby:
Riordan's Tavern (sit at the bar)
The Palm (probably not a quick dinner)
Zucca (I've done the happy hour, that worked pretty well in terms of time and food)
Roy's (sit at the bar), this would be the best food option IMO.
I've resorted to eating some sort of tie-over snack in Staples, and then getting a good dinner later in Jtown or Chinatown.