Quantum field theory can be formulated in curved spacetime, so in that sense GR and QM can "work" together. The problem is that such a theory cannot be fundamental; it must be an approximation to some deeper theory. The reason, heuristically, is that, according to QM, if a system has different possible configurations, there should be an amplitude for it to be in each of them. But GR doesn't work like that: it doesn't say there are amplitudes for different possible spacetime geometries, stress-energy tensors, etc. It says there is only one spacetime geometry, stress-energy tensor, etc.--i.e., only one configuration for the system. (This is true of any classical, i.e., non-quantum, theory.)
So any situation in which quantum effects could create different possible spacetime geometries, stress-energy tensors, etc., with amplitudes for each, cannot be properly modeled using GR. But such situations should occur whenever any significant quantity of mass or energy is subjected to quantum superposition. A typical such situation is determining how a mass moves by some quantum process, such as the decay of a radioactive atom.
Saying that gravity is "not a force" in the sense of Newtonian physics (but instead is spacetime geometry) is not the same as saying that gravity is not an "interaction" in the sense of quantum field theory. The latter statement could very well be true; and in fact, the simplest approach to quantum gravity is to assume it is true, and to model gravity at the fundamental level using a massless, spin-2 quantum field, for which the "graviton" is the corresponding particle excitation. It was shown in the 1960s and 1970s that the classical limit of such a theory is in fact GR itself--i.e., that the classical limit of the field equation satisfied by such a quantum field is in fact the Einstein Field Equation of GR. So such a theory of quantum gravity is perfectly consistent with GR.
The problem with this theory of quantum gravity is that it is not renormalizable, which basically means that it is not expected to be a complete theory by itself. There should be extra terms in the theory describing extra interactions which only occur at extremely high energies, and which we have no way of probing experimentally. In fact we have no way of even probing the quantum nature of the spin-2 field itself experimentally, since such quantum aspects are not expected to become significant until we reach length scales on the order of the Planck length, some 20 orders of magnitude smaller than the smallest length scales we can currently probe.
We don't in any practical sense; as I noted above, the quantum aspects of gravity are much too small for us to probe now or in the foreseeable future. We care because we know, for the reasons given above, that our current theories are not complete, and we would like to try to find more complete theories.
