It is said that we can't study quantum gravity because gravity is a weak force. But gravity and acceleration are the same. Why can't we study quantum gravity in a strongly accelerated frame of reference?
This is a classic case of equivocation. The word "gravity" is used in the question with two related-but-different meanings:
When we say "gravity is weak," we mean that a lot of stuff (like matter) is needed in order to give spacetime a noticeable amount of curvature.
When we say "gravity and acceleration are the same," we mean that the experience of standing on the surface of the earth is locally indistinguishable from the experience of accelerating (being pushed from your feet toward your head) in flat spacetime. Conversely, the feeling of weightlessness you have when drifting in deep space is the same as the feeling of freely falling toward the surface of the earth with no resistance at all.
Distinguishing between those two meanings of the word "gravity" is the key to answering the question. We have always known (since before I was born, at least) how to formulate quantum physics in a prescribed spacetime background, with or without curvature, and with or without acceleration. That's how Hawking radiation was originally derived: using a prescribed spacetime background. When people say that quantum gravity is difficult, that's not what they mean. The difficult part is accounting for the fact that quantum stuff causes spacetime curvature. Things like Hawking radiation and the Unruh effect are about what acceleration and/or spacetime curvature does to quantum stuff. The hard part of quantum gravity is understanding the converse: what quantum stuff does to spacetime curvature.
Gravity and acceleration are not the same. Locally, (meaning in a sufficiently small region of space and time), it's not possible to distinguish the effects of uniform acceleration and gravity, due to the equivalence principle. But over a sufficiently large distance, you can tell the difference. For example, two people on opposite sides of the Earth are pulled in opposite directions (both toward the center); you cannot explain this by one accelerating reference frame, although you can define two local frames near each observer.
You can study quantum field theory in an accelerated reference frame; one of the classic results is that a state which an inertial observer perceives to be the vacuum (no particles), is seen by the accelerated observer to have a thermal bath of particles. This is known as the Unruh effect.
By the equivalence principle, you can use the Unruh effect to understand Hawking radiation locally near the event horizon of a black hole. The infalling observer sees a vacuum state, while an observer remaining at a fixed distance near but outside the black hole sees a thermal bath of radiation, because they are accelerating relative to the infalling observer. The Unruh effect is not sufficient to say what an observer asymptotically far away from the black hole will see, but a full calculation shows that the radiation seen by an observer near the event horizon propagates to infinity (after being redshifted) and is perceived as Hawking radiation. (At least, in the "effective field theory of gravity", but a full quantum theory of gravity might change the story).