I dropped out of physics before getting my 4 year degree, so I was just barely starting to learn quantum, but from what I've read (and bear in mind I am far from an expert), if you combine QM with anything that includes gravity, the resulting mess starts spitting out infinities where finite values are expected.
And that kind of thing tends to make physicists very cranky.
Pretty much it.
Actually, such infinities emerge more generally, but in the Standard Model, at least, they can be subtracted out, a process called renormalization. Troublesome infinities started to appear as physicists started to do quantum mechanics on elementary-particle fields, starting with photon and electron fields. But they eventually learned how to redefine masses, charges, and field strengths so as to absorb these infinities. The first successes were in the 1950's, with photons and electrons, giving quantum electrodynamics. Richard Feynman, Shinichiro Tomonaga, and Julian Schwinger won a Nobel Prize for that work in 1965. This work was easily extended to other particles' electromagnetic interactions, though strong and weak ones were more difficult.
In the late 1960's and early 1970's, physicists showed that a kind of theory called a "nonabelian gauge theory" is also renormalizable. Gerardus 't Hooft and Martinus Veltman got a Nobel Prize for that work in 1999. A nonabelian gauge theory is a theory much like electromagnetism, but with more than one "charge", and with the "photons" having some charge-anticharge combinations. This makes the "photons" interact with each other, thus producing the "nonabelian" feature.
This made it possible to construct well-behaved theories of the strong and electroweak interactions, with the electroweak interaction being a combination of the electromagnetic and weak ones. Thus giving the Standard Model of particle physics.
But if one tries to renormalize a quantized version of general relativity, one gets into trouble. One might try fixing it by adding more particles in suitable ways, but one is unable to stop. The closest that we have gotten to a well-behaved quantum theory of gravity has been string theory, and that works by having an infinite number of string modes. But string theory has plenty of problems of its own. One can get an approximation of the Standard Model out of it, but there are oodles of other possible solutions, and it is not very clear why our Universe would have one of them and not some other. Is it a sort of anthropic natural selection? That's because we'd see only a kind of Universe that allows us to come into existence. So there could be a string-solution multiverse with oodles of sterile string-solution universes.