Although in general physicists believe a lot of things from spacetime for QM, like that information can't travel faster than light, it isn't actually baked into QM.
That's my understanding at least.
There are several textbooks[1] covering extensions of this approach to curved spacetimes generally, and one will encounter the https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation#... in graduate school settings. Quantum mechanics on some specific curved spacetimes are exactly Klein-Gordon solvable.
The Standard Model of Particle Physics incorporates the Poincaré Group, which is the group of Minkowski spacetime ("flat spacetime") isometries. This means that the Standard Model particles are the same independent of where and when they are in empty flat spacetime (3 spacelike and 1 timelike translation invariances), their orientation (rotational invariance about the three spacelike axes), and they transform reliably under boosts (differences in constant velocity along the spacelike axes). So the Standard Model is defined by flat spacetime: and the causal structure of flat spacetime (in which the constant c plays a key role) is very much baked in. Studying modern quantum mechanics mostly means looking at patches of effectively flat spacetime in which there are particles of the Standard Model occupying different locations in the patch, boosted relative to each other, and interacting via the other mechanisms captured in the Standard Model's formulation.
We inhabit a type of curved spacetime in which General Relativity guarantees at least an infinitesimal patch of exactly flat spacetime around every point in the universe. In regions with only gentle curvature -- like laboratories on the surface of the Earth, or in space probes in the solar system -- any curvature corrections to the assumed exactly flat spacetime of the Standard Model are tiny, because the region of effective (rather than exact) flatness is very large compared systems of particles under experimental study of their quantum behaviour. "Pretend it's flat" works exceedingly well in practice. When a researcher has to consider curvature (e.g. in relativistic massive objects like neutron stars) she or he can continue to work perturbatively against a flat-by-definition spacetime.
The problems arise in the difference between a guarantee of a microscopic region of flat spacetime and an infinitesimal region of flat spacetime: if the radius of curvature is small compared to the spatial extent of a particle, things get ugly quickly, especially as we take the wavelength of the particle smaller (which means the energy of the particle climbs, and that is the sort of energy which creates spacetime curvature, so we get a nonlinear feedback, and classical General Relativity and Quantum Field Theory in Curved Spacetime make annoyingly different and incompatible predictions about what happens as one takes the limits of high particle energy and high curvature. Fortunately this incompatibility seems likely to occur only hidden inside event horizons, so it is a problem for the theories rather than a practical problem for all of us who are not actually rapidly approaching death within a black hole -- there also may be consequences for as-yet-undiscovered ancient tiny black holes, or stellar black holes many trillions of trillions of years in our future, so again we can take our time to understand the theoretical conflict).
A bit more technically, the problem arises in perturbative approaches to QFT on curved spacetime: at low energies and low curvatures we have a fairly small number of correction terms which can be written out as a https://en.wikipedia.org/wiki/Taylor_series which we can truncate because the higher-order terms are demonstrably irrelevant. As we increase energies and curvature, irrelevant terms become marginal, then relevant; additionally, we start having to add more non-irrelevant terms. https://en.wikipedia.org/wiki/Renormalization allows us to squash some of these terms together, but eventually we get an overwhelming growth of non-ignorable corrective terms and lose the ability to make predictions using this approach.
This breakdown in perturbative renormalization ("perturbative quantum gravity" [2][3]) gives a useful qualitative definition of "strong gravity": it's where the perturbative approach breaks down. In terms of Feynman diagrams, it's where loops of gravitons enter into the picture; a bit more colloquially, it's where "gravitation's self-gravitation starts becoming non-ignorable".
Although not a dead area of research, looking for ways to make renormalization work for strong gravity is less fashionable than looking for non-perturbative quantum gravity that (a) matches perturbative quantum gravity right up to the weak-side boundary of strong gravity, including classical General Relativity in weak gravity, (b) is calculable in practice, and (c) solves other non-gravitational problems that plague high energy particle physics that are amenable to testing, since we probably can't extract observational or experimental data from regions of strong gravity.
Additionally, classical General Relativity fairly generically predicts the presence of gravitational singularities in spacetimes with significant amounts of matter[4]. Such singularities destroy the total predictability of the entire spacetime from a total sample of all the variables on an arbitrary slice across the whole space at a given time coordinate. In other words, there's an incompatibility between classical General Relativity and traditional initial values surfaces approaches to solving physics problems. This problem worsens in the presence of Quantum Fields because of Hawking Radiation: instead of a literal singularity there is instead a trapping structure that evaporates (or mostly evaporates, in "remnant" proposals) in the far future of most black hole systems. But the matter that is released in the evaporation cannot be predicted from the matter that was thrown into the black hole before evaporation, and at late times we lose the ability to account for quantum entanglements that existed when the black hole was growing. Unfortunately there are numerous black hole candidates in our universe, which we also know is filled by matter representable by quantum fields. Although this is not strictly an incompatibility of General Relativity and Quantum Field Theory, quantum physicists are very keen on preserving https://en.wikipedia.org/wiki/Unitarity_(physics) which is lost in black hole evaporation as is understood today. Preserving unitarity in the presence of strong gravity is a fourth goal of modern research into quantum gravity.
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[1] e.g. Parker & Toms, QFT in Curved Spacetime (2009), https://books.google.co.uk/books?id=5nNuGMBBTjMC&dq=L.+Parke...
[2] http://www.staff.science.uu.nl/~hooft101/lectures/erice02.pd...
[3] https://en.wikipedia.org/wiki/Canonical_quantum_gravity
[4] https://en.wikipedia.org/wiki/Raychaudhuri_equation is a good place to start