quote (v.) Look up quote at Dictionary.com
late 14c., coten, "to mark (a book) with chapter numbers or marginal references," from Old French coter, from Medieval Latin quotare "distinguish by numbers, number chapters," from Latin quotus "which in order? what number (in sequence)?," from quot "how many," from PIE *kwo-ti-, from pronomial root *kwo- (see who).

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). Modern spelling with qu- is from early 15c. The business sense of "to state the price of a commodity" (1866) revives the etymological meaning. Related: Quoted; quoting.
quote (n.) Look up quote at Dictionary.com
"a quotation," 1885, from quote (v.). From c.1600 as "a marginal reference." Quotes for "quotation marks" is from 1869.