During the Tournament in honour of Ned Starks appointment to Hand of the King (The Tourney of The Hand), Ser Gregor kills Ser Hugh of the Vale in what seems to be an accident.
However, both Ned Stark and Ser Gregor's brother - Sandor "The Hound" Clegane - seem to think that this was not an accident, and we are also lead to believe that it is not. Ned Stark seems to think that Ser Hugh was killed because Ned had wanted to question him.
However, in retrospect (ASOS), we know that it was not Cersei who killed Jon Arryn, we only know that she suspected that Jon Arryn and Stannis knew the secret of who really fathered her children.
I can imagine that Cersei would possibly have had Ser Hugh killed, just in case he actually knew something that she suspected Jon Arryn or Stannis might have suspected, but why would she do it at that time? And why use Ser Gregor in that case?
Looking back at it now, it seems that it is more likely that this was purely coincidental. If Cersei had suspected that Ser Hugh knew anything, she would have had him killed right away. And frankly, it could have been done at any time, and not at the tournament in front of half of King's Landing, done by her own father's bannerman.
What was the reason for Ser Gregor killing Ser Hugh?
Anything is possible, but I prefer having some evidence to support speculation. My personal opinion is that Ser Gregor would not take killing orders from just anyone. The man is not an idiot. It would have to be a Lannister, or someone like Pycelle who is a Lannister loyalist.
It seems that GRRM has left this particular event unexplained, when so many other events were actually explained. For example, we heard who sent the assassin to kill Bran, we have some clue as to what Ned promised Lyanna, we know who poisoned Jon Arryn and who gave the order, we know that Cersei tried to have all of Robert's bastards killed, etc. But no hint as to why this incident happened, besides the obvious?