This is a final table ICM problem.
Although QQ is a very good hand, the consequences of losing are severe on the final table. Your hand needs to be a huge favorite over his range if you go allin, because your loss in equity is very high if you lose and your gain is much smaller than your loss if you win (-77% vs +45%)
(I looked up the stars tourney payout schedule and ante schedule for the numbers, assuming 2.20 buyin with 1500 entrants)
1st 540.00
2nd 324.00
3rd 192.00
4th 162.00
5th 132.00
6th 102.00
7th 78.00
8th 59.40
9th 39.30
Here is a range that would be +EV to go allin against. If his range would be narrower, then theoretically, going allin would be -EV
equity win tie pots won pots tied
Hand 0: 64.035% 63.23% 00.80% 98528580 1250781.00 { QdQh }
Hand 1: 35.965% 35.16% 00.80% 54789522 1250781.00 { 77+, AJs+, KQs, AJo+, KQo }
I am not giving recommendations on how to play this hand. I am just pointing out that final table play is not the same as earlier in the tourney. One alternative might be to flat call and bet if he checks the flop (even if there is an A or K on the flop), fold if he bets a flop with an A or K, and obviously move in on any flop without an A or K.
The same kind of analysis could be done for his hand from his perspective, once he raises with AK and gets moved in on. Unfortunately, I don't have time to do that analysis, but the results might be contrary to what we expect, using ICM.