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.
Old English hof "hoof," from Proto-Germanic *hōfaz (source also of Old Saxon, Old Frisian hof, Old Norse hofr, Danish hov, Dutch hoef, German Huf "hof"), perhaps from PIE *kop- "to beat, strike" (source also of Sanskrit saphah "hoof," Polish kopyto "hoof;" see hatchet (n.)). But Boutkan acknowledges only Indo-Iranian cognates and writes, "We may be dealing with a typical relic form that only survived in the periphery of the IE area ...." For spelling, see hood (n.1).
A hoof differs from a nail or claw only in being blunt and large enough to inclose the end of the limb; and almost every gradation is to be found between such structures as the human nails, or the claws of a cat, and the hoofs of a horse or an ox. The substance is the same in any case, and the same as horn, being modified and greatly thickened cuticle or epidermis. [Century Dictionary]
Hoof-and-mouth disease is attested from 1866. Phrase on the hoof is from 1750 as "walking;" later it was cattlemen and butchers' slang for "not (yet) slaughtered."