In the end: do gravitational waves implies the existence of gravitons? Or it's something unrelated and off topic?
The concept of gravitons arises from a quantized gravity, and quantization of gravity is still an effective theory since it has infinities that have not been renormalizable or containable in some sense in the effective theories used on trust, that the matter will be settled.
String theories do have mathematically consistent quantization of gravity, and gravitons, but they are not at a research point to propose a standard model that will describe all existing data and predict new phenomena.
Thus the graviton is an expected new particle, but even its theoretical existence is precarious, until a definitive quantization of gravity.
Experimentally, when the BICEP2 experiment was thought to have detected gravitational waves from the imprint of their polarization on the cosmic microwave background, the footprint of gravitons , i.e. quantized energy transfer, was thought to have been detected.
Craig Hogan, director of the Fermilab Center for Particle Astrophysics, told physicsworld.com that "If it's confirmed, it is truly profound – the first direct evidence not only for inflation, but of a quantum behaviour of space and time. The image of polarization is a relic imprint of roughly a single quanta of graviton action."
Unfortunately the signal could not be separated from scatterings off space dust. A new experiment with higher resolution, BICEP3 has started data taking and we have to wait and see.