In addition to Hugo's fine answer here are some more details: If you look at etymology you'll find
O.E. cwic "living, alive," from P.Gmc. *kwikwaz (cf. O.Fris. quik, O.N. kvikr "living, alive," O.H.G. quec "lively," Ger. keck "bold"), from PIE base *gwiwo- "to live" (see bio-). Sense of "lively, swift" developed by c.1300, on notion of "full of life."
c.1000, "part of speech," from O.Fr. part, from L. partem (nom. pars, gen. partis) "part, piece, side, share," related to L. portio "share, portion," from PIE base *per- "to assign, allot" (cf. Gk. peprotai "it has been granted," Skt. purtam "reward," Hittite parshiya- "fraction, part").
In usage you would say a person of quick parts.
Combining the meaning and usage it is plausible that the original meaning was "lively (witty, eloquent) speech (responses, oratory skills)".
Examining books on usage I found Webster 1832, treat it like this
4. Active ; brisk ; nimble ; prompt ; ready. He is remarkably quick in his motions. He is a man of quick parts
However that does not seem to shed the light on it, perhaps this passage shows a more detailed interpretation
He is grave with the grave, and lively with the gay ; to the man of quick parts he is quick and penetrating
The use of the idiom seems to be steadily declining over the past two centuries and is rarely used these days.