"Buck" as slang for one dollar is of unknown origin though it dates to the mid-19th century.

It's possible that it comes from the early to mid 18th century in America when buckskins were considered as legal tender in many parts of America.

For example the trader Conrad Weiser kept a journal of his travels in 1748 as he went and traded with the native populations. At one point he gave a speech and talked about a fellow merchant "He has been robbed of the value of 300 bucks", and in another entry his speech is regarding the price of whiskey "Your Brethren, therefore have order'd that every cask of Whiskey shall be sold to You for 5 Bucks in your Town . . ."

The Online Etymology Dictionary says this about "quid":

"one pound sterling," 1680s, British slang, possibly from quid "that which is, essence," (c.1600, see quiddity), as used in quid pro quo (q.v.), or directly from Latin quid "what, something, anything." Compare French quibus, noted in Barrêre's dictionary of French argot (1889) for "money, cash," said to be short for quibus fiunt omnia.