late 14c., coten, "to mark or annotate (a book) with chapter numbers or marginal references" (a sense now obsolete), from Old French coter and directly from Medieval Latin quotare "distinguish by numbers, mark off into chapters and verses," from Latin quotus "which in order? what number (in sequence)?," from quot "how many," from PIE *kwo-ti-, from pronominal root *kwo-.
The sense development is via "to give as a reference, to cite as an authority" (1570s) to "to copy out or repeat exact words" (1670s), in writing or printing, "inclose within quotation marks." In Middle English also "to compute, reckon." The modern spelling with qu- is attested from early 15c. The business sense of "to state the price of a commodity" (1866) revives the etymological meaning. Also see unquote. Related: Quoted; quoting.