Not a physicist.
Assuming gravitons exist, then that would mean that they are constantly being exchanged between objects of mass. This would imply that objects containing mass at a fixed distance would spontaneously become entangled due to their gravitational interaction.
A bit of googling reveals [0], which I believe describes such an experiment.
(As with electrons and photons, in a lot of cases their individuality isn't meaningful, so we talk of light wavefronts and bandgaps and so on)