Calque of Latinpomum Adami, which is found in the botanical sense from 1560 and the anatomical sense from 1600. According to the earliest sources, the anatomical usage derives from a common belief that Adam was punished by God for the fall of man by having a piece of the forbidden fruit lodged in his throat (see 1662 citation).
1662 , Nicholas Culpeper and Abdiah Cole, transl., Bartholinus Anatomy, London: Peter Cole, translation of Protuberantia illa in collo anterius conspicua, dicitur Pomum Adami; [quia vulgo persuasum in Adami faucibus pomi fatalis partem ex pœna Divina remansisse, & ad posteros translatam] by Thomas Bartholin, page 123:
That same bunch which is seen on the foreside of the Neck, is called Adams Apple, because the common people have a beliefe, that by the judgement of God, a part of that fatal Apple, abode sticking in Adams Throat, and is so communicated to his posterity
1597, Gerarde, John, The Herball or Generall Historie of Plantes, London: John Norton, page 1281:
The fourth [name of the citron] is named of diuers Pomum Assyrium, or Citron of Assyria, and may be Englished Adams apple, after the Italian name, and among the vulgar sort of Italians Lomie, of whom it is also called Pomum Adami, or Adams apple, and that came by the opinion of the commom rude people, who thinke it to be the same Apple, of which Adam did eate in Paradise when he transgressed Gods commandement