To the best of my knowledge no on-screen statement has been made conclusively either way, and Doctor Who 'canon' only exists inasmuch as sometimes it doesn't actively contradict itself. But we can extrapolate that the most likely scenario is:
River Song should have one heart.
She's a human, not a Gallifreyan, and that makes all the difference. Being a Time Lord shouldn't have any impact on gross biology like the quantity of one's organs, because "Time Lord" is not a race, nor a species. It just means you've been subjected to a suite of (poorly-defined) biotechnological enhancements invented by Gallifreyans--and almost always only used on Gallifreyans, so Time Lords have two hearts because Time Lords are almost exclusively Gallifreyan, not because being a Time Lord does that to you. (See the Doctor's daughter, who seems to probably be a Gallifreyan and not a Time Lord but had two hearts anyway.)
River Song is a human, born of human parents, who has "Time Lord DNA" upgrades/enhancement/modifications from TARDIS/Time Vortex technobabble [waves hand vigorously]. She's displayed Time Lord features like regeneration because of this (but there are many indications she didn't get the full suite of enhancements). There's no reason, however, that she should have two hearts: that seems to be a quirk of Gallifreyan anatomy and River Song is not Gallifreyan.
The idea of human Time Lords, by the way, is not new.