The story
Because U.S. dollar bills are green; the term reflects euphemistic irony, as it implies that a handshake that hands them off is more than a mere handshake.
Etymology from Wiktionary, CC BY-SA 4.0.
green handshake
Today it means: A bribe, or a transfer of money that the interested parties represent as a tip but that disinterested third parties are likely to view as a bribe. Full definition →