I think the answer is almost certainly no, and is mostly unrelated to how smart LLMs can get. The issue is that any theory of quantum gravity would only be testable with equipment that is much, much more complex than what we have today. So even if the AI came up with some beautifully simple theory, testing that its predictions are correct is still not going to be feasible for a very long time.

Now, it is possible that it could come up with some theory that is radically different from current theories, where quantum gravity arises very naturally, and that fits all of the other predictions of of the current theories that we can measure - so we would have good reasons to believe the new theory and consider quantum gravity probably solved. But it's literally impossible to predict whether such a theory even exists, that is not mathematically equivalent to QM/QFT but still matches all confirmed predictions.

Additionally, nothing in AI tech so far predicts that current approaches should be any good at this type of task. The only tasks where AI has truly excelled at are extremely well defined problems where there is a huge but finite search space; and where partial solutions are easy to grade. Image recognition, game playing, text translation are the great successes of AI. And performance drops sharply with the uncertainty in the space, and with the difficulty of judging a partial solution.

Finding physical theories is nothing like any of these problems. The search space is literally infinite, partial solutions are almost impossible to judge, and even judging whether a complete solution is good or not is extremely difficult. Sure, you can check if it's mathematically coherent, but that tells you nothing about whether it describes the physical world correctly. And there are plenty of good physical theories that aren't fully formally proven, or weren't at the time they were invented - so mathematical rigour isn't even a very strong signal (e.g. Newton's infinitesimal calculus wasn't considerered sound until the 1900s or something, by which time his theories had long since been rewritten in other terms; the Dirac delta wasn't given a precise mathematical definition until much later than it's uses; and I think QFT still uses some iffy math even today).