^ first of all, this is way beyond my level of physics understanding. I'm only in senior year, so I wont know for sure for a few more.
Basically from what I understand, is that you can do relativistic quantum mechanics in curved space (in fact, that's what hawking radiation is all about). The problem is that quantum field theory, which is kind of like the "grown up" quantum mechanics, and treats particles as excited states of an underlying field. They developed quantum field theory because the single-particle dirac equation, which is basically like like the relativistic shrodinger equation, gave lots of bad results, such as negative energy and things. If you formulate the dirac equation as a field theory, then a lot of those problems are solved.
They basically found a process called "renormalization" that allowed them to make corrections to the fields that got rid of infinities, which physically do not make sense.
The problem with GR and QFT is when you try to quantize the gravitational field. It turns out that the field is nonrenormalizable. You would basically have to give it infinitely many "bandages" to make it work. That's kind of what guys are trying to do with string theory, they are trying to find the paramaters that would make the gravitational field renormalizable.
I think that's correct, but keep in mind, all of this info has come from like brian green and steven hawking books, or talking to grad students/profs in a casual setting. I have only taken a few QM courses so far.