No. The higgs is a field that gives some elementary particles themselves (the W and Z bosons) mass, but doesn't necessarily say anything about gravity or how gravitic 'force' is transferred.

There was a lot of media hype about 'the god particle' that doesn't really translate into reality. I've said this in another comment, but if you add up the mass of the constituent quarks of a neutron, you get approximately 1% of a neutron's mass. The majority of the mass comes from interactions with strong nuclear force which are mediated by gluons, which are themselves massless.

There is no current agreed upon understanding of quantum gravity or if gravitons exist. I think the big contenders right now are String Theory (which seems to be having issues progressing in a way that is useful) and loop quantum gravity, but there are a lot more theories than that.