Pathfinder Adventure Path, Starfinder Adventure Path, Starfinder Roleplaying Game Subscriber
Hmm... I think I would rule with the monk on this one. The haste spell allows him to make an extra attack when taking a full attack action. The monk ability essentially adds an extra attack to the flurry of blows ability ("can make one additional attack at his highest attack bonus when making a flurry of blows attack"). So one effect changes flurry of blows, one effect changes the full attack action so I don't see why they wouldn't stack.
The problem is that flurry of blows is in fact a full-round action.
Not according to the PRD:
Flurry of Blows (Ex): Starting at 1st level, a monk can make a flurry of blows as a full-attack action