From Longman Dictionary of Contemporary Englishbill and coobill and cooLOVE old-fashioned to kiss and talk softly → bill
Examples from the Corpus
bill and coo• With wheedling beaks, her brilliant needles bill and coo.• In August, however, she and Andrew were billing and cooing about each other in Hello! magazine.• One is the Prime Minister, who now bills and coos where she used to bellow.• Immediately the pigeons began to strut and flutter out, billing and cooing with pleasure. 