The Defense Professor sent Harry the forged message after 11:04pm, but early enough that Harry was able to arrive at 6:49 with 1 hour left on his time turner. The time-turner has been confiscated, but this is still a big security hole.
Example possibility: Harry and Quirrel arrive at the mirror, and time-turned Harry is already there with Dumbledore, Moody, Madam Bones and a dementor, which had not been detected by present Harry because patronus 2.0 is out and shielding everyone from it. Quirrel can’t apparate away in Hogwarts, patronus 2.0 stops any AK, and Dumbledore & friends can hold Voldemort off long enough for the dementor to start sucking face with Voldemort, who is particularly vulnerable. Harry retrieves his time-turner and goes back an hour to message Dumbledore via patronus and set this ambush up. Asking Harry to state whether he has betrayed him is of limited use here, since the betrayal can happen in the future but impact the present.
Why has The Defense Professor deliberately preserved an hour on Harry’s time-turner? It must be significant to his plan.
I think it’s stronger than that. Quirrell suggesting this memory can only have two implications that I can see:
Either 1) A baby remembering an event like this in the HPMOR universe is not, in any way that Quirrell would notice, unusual, and would even be expected. This seems unlikely and unsatisfying to me. No reason for the existence of this discrepancy between their world and ours has been shown, no excuse made. I also think “how does anyone know this story?” is the nagging rationalist doubt Harry experiences when he first hears of it. Especially—how would one remember a conversation in detail before having learned the language it is in, or any language at all? If anything he would remember tones, emotions, sights. His supposed recall is an extremely improbable feat for baby Harry.
Or 2) Quirrell knows both that the memory was implanted and its contents. This is consistent with Quirrell, particularly that he has made the right guess for the wrong reasons, as previously described in Ch.49, “Prior Information”. The main lesson from that chapter is that we should not assume that Quirrell’s (or anyone’s) seemingly extraordinary deductions are actually deductions, but look for other reasons why he would have that information.
The simplest explanation for #2 is that Quirrell knows because Quirrell placed the memories. I am assuming that the inability of Quirrell to interact magically with Harry is real because EY has called it a literary necessity, but Quirrel/Voldemort/Riddle/Monroe could have just placed the (unusually clear and detailed) memory into the infant Harry before turning him into a horcrux that he would then not be able to magically interact with (is this a thing that always happens with human horcruxes?) There would be no need to obliviate the real events afterwards, the 15-month-old babe would not remember anyway, but he could also just have Bellatrix do it. This would be part of some version of “Voldemort faked his own death.” There are a few of versions of how this might have happened, and I haven’t settled out what I believe yet.
The other possibility for #2 is that someone else, most likely Dumbledore, placed the false memories for reasons that are not clear to me and that Quirrell has come by this information somehow. If he had done it, and Minerva knew the secret she might be a weak occlumency link in the Order’s chain, allowing Voldemort’s legendary legilimency to obtain any information she has. She seems highly trustworthy but not particularly secure. I can’t think of any reasons why Dumbledore would do this, though. It seems to only disadvantage Dumbledore’s priorities.
It is possible, I suppose, that dementors have the ability to make their victims recall even the faintest of bad memories in stunning detail, or that Dumbledore has some inscrutable reason for faking the events of that night. Overall I think the most likely explanation is that Quirrell planted Harry’s memory of Godric’s Hollow.