Quirrell's attempt on Harry's life, trying to throw him off his broom during a school Quidditch match, doesn't seem to fit with Voldemort's style or plans to me. If my understanding of the chronology is correct, Voldemort was already sharing Quirrell's body. Did Voldemort instruct Quirrell to kill Harry, getting rid of him rather easily and unceremoniously, or did Quirrell simply try to kill his ally's enemy in a misguided way?
3 Answers
Probably
It seems likely that Voldemort must at least have approved of what Quirrell was doing. To start with, Quirrell lacks much gumption or initiative:
“Sometimes,” he said, “I find it hard to follow my master’s instructions —he is a great wizard and I am weak—”
Harry Potter and the Sorcerer’s Stone
Voldemort specifically decided to possess Quirrell after his failure to procure the Stone in order to ensure his orders were carried out properly:
“When I failed to steal the stone from Gringotts, he was most displeased. He punished me… decided he would have to keep a closer watch on me....”
Harry Potter and the Sorcerer’s Stone
It seems unlikely, given this, that he would have tolerated Quirrell carrying out plans to kill Harry of which he did not approve.
As to whether it was out of character for Voldemort to try to kill Harry “unceremoniously,” the answer is no. The only time Voldemort actually tried to kill Harry with ceremony, as it were, was in Goblet of Fire, with the main aim of enhancing his power through the use of Harry’s blood, and proving to his Death Eaters that Harry was nothing, thus reinforcing their loyalty.
The rest of the time, he hardly stands on ceremony: in Order of the Phoenix, he casts Avada Kedavra about 10 seconds after he appears; he similarly does not hold back several times in Deathly Hallows. As Riddle, he does gloat a bit in Chamber of Secrets, I suppose. His main fixation, though, is killing Harry personally, and while it may not have been so strong back in the first book, it hardly matters: with him controlling Quirrell and directing his actions, he probably would have seen it as the same thing.
-
Plus, when Quirrell tries to kill Harry at the Quidditch match, Voldemort has not been defeated (knowingly) by Harry. He still doesn't know why Harry survived as a baby, but unlike CoS onwards, Harry hasn't yet stood up to him, fought him, and lived to tell the tale again—so it's not personal yet in the same way as it is later on. Commented Sep 17, 2016 at 22:17
-
I would say the battle in the Chamber of Secrets is quite a ceremony to me, even though there was no-one to witness the events. Commented Sep 22, 2016 at 21:15
It's hard to know whether Voldemort explicitly ordered it of course, but he certainly had more than enough time to ask Quirrell to stop mucking about during the match.
As for the whys and wherefores, Quirrell does give a reason for it. I think there's a little bit more to it than getting rid of Harry because he was Harry. Harry had proven himself a bit of a threat.
'You're too nosy to live, Potter. Scurrying around the school at Hallowe'en like that, for all I knew you'd seen me coming to look at what was guarding the Stone.'
Harry Potter and the Philosopher's Stone - pp.209-10 - Bloomsbury - Chapter 17, The Man with Two Faces
It's hard to know exactly how the conversation between Quirrell and the back of his own head went, but it seems like him himself and he came to the conclusion that they needed Harry out of the way sooner rather than later.
It does sound a bit like it was Quirrell who came to this conclusion, but at the very least, Voldemort seems a-okay with it.
There's no reason to forbid Quirrell from offing Harry, unless the attempt fails in a way that exposes Voldemort's existence.
Either Harry dies or Voldemort gains intel on Harry's defenses. So it's a win-wind for him.