There are gravitational wave observations that can test General Relativity (GR), looking for disagreement with classical predictions. There are some particular disagreements that are motivated by Loop Quantum Gravity and/or string theory, and there may also be generic deviations be from classical GR that don't point to a particular theory.
Classical GR predicts gravitational waves (GW) propagate at the speed of light and have no dispersion. It also predicts no monopolar or dipolar radiation. Another way of saying this is that there is no scalar or vector gravitational radiation, only tensor gravitational radiation.
There are specific observations that could test these predictions for example Larson and Hiscock proposed using binary white dwarfs to test the speed of GWs. This puts effective bounds on the mass of a graviton (predicted massless). This observation requires low frequency GWs that LIGO cannot observe. A space based detector like eLISA would be required.
There are several alternative theories of gravity that are extensions of GR. For example Brans-Dicke gravity is a "scalar-tensor" theory. There are also "scalar-vector-tensor" theories. These would drastically modify the form of gravitational waves. So much so, that there are limits on these theories based on relatively weak-field binary pulsar observations.
GWs also encode information about the trajectories of the particles producing them. There is plenty of active research looking for how to detect generic deviations from GR in the motion of GW producing particles, for example LIGO's TIGER project and the Parameterized Post-Einsteinian framework. These methods would require a very, very loud GW signal or many GW detections to work effectively, making these sorts of tests longterm goals of GW astronomy.
Currently there are no observed deviations from classical GR. Any observed deviation would need to be addressed in the next generation of gravitational theory. While these methods don't specifically test Quantum Gravity, they may motivate advances in theory.