I am not sure what sort of answer you expect. For a massive particle, the minimum possible energy it can possesses is its rest mass, when the particle has zero velocity (if we ignore the problems associated with such states in quantum mechanics). But the graviton is expected to be massless. One might be tempted to say that the answer would be zero. This unfortunately poses a problem: a graviton of zero energy has no energy, momentum, or ability to interact with anything else. Such a 'graviton' would be a complete dud and it is not very productive to ascribe physical existence to such things. Mathematics tells us that "the smallest number greater than, but not equal to zero" cannot be defined. So there is no answer to your question: a graviton can have arbitrarily small energy, but (IMO) not zero. There is no relation(*) that fixes a minimum energy or a quantum scale for it (contrast, for example, angular momentum which can be proven in QM to be quantized in increments of h-bar/2).
This question is similar to "what is the minimum energy of a photon?", which is equally undefined.
(*): There is no agreed upon theory of quantum gravity and it is possible that some such theories may allow for that.
