Cough
Middle English: of imitative origin; related to Dutch kuchen ‘to cough’ and German keuchen ‘to pant’.
wiktionary
From Middle English coughen, coghen, from Old English cohhian (compare Old English cohhetan(“to shout”)), from Proto-Germanic *kuh-(“to cough”). Cognate with Dutch kuchen(“to cough”), German keuchen(“to pant”), Albanian hukat(“pant, gasp”).
etymonline
cough (v.)
"a violent, noisy effort to expel air from the lungs," early 14c., coughen, probably in Old English but not recorded, from Proto-Germanic *kokh-(source also of Middle Dutch kochen, Middle High German kuchen), with the rough "kh" of German or of Scottish loch. Onomatopoeic. Related: Coughed; coughing.
As a noun from c. 1300, "single act of coughing." As "illness or other condition that affects the sufferer with frequent coughs or fits of coughing," by 1742. Cough-drops attested by 1829; cough-medicine by 1828. To cough up "to present, hand over" is from 1894.
