For to pony up, etymonline.com says
1824, in pony up "to pay," said to be from slang use of L. legem pone to mean "money" (first recorded 16c.), because this was the title of the Psalm for March 25, a Quarter Day and the first payday of the year (the Psalm's first line is Legem pone michi domine viam iustificacionum "Teach me, O Lord, the ways of thy statutes").
(which in itself is one awesome etymology if true)
For the more rare, although in use everywhere from Huckleberry Finn to modern San Francisco Chronicle, to pungle (up), Merriam-Webster says,
"Pungle" is from the Spanish word "pongale," meaning "put it down," which itself is from "poner," meaning "to put" or "to place," or more specifically "to contribute money." The earliest uses of "pungle" are from the 1850s
But does that Spanish poner as in "to contribute money" come from the same slang Latin pone?
Best Answer
[Edited: The Oxford English Dictionary says that to pony up simply comes from pony, so Colin informed us. That means Etymonline's etymology is highly controversial. Edited again: The noun pony has been used to mean "£25" since at least 1797, according to the OED. To pony up has been used to mean "to pay up" at least since 1824 in American slang, says the same dictionary. Is there a connection between English and American slang?]
Assuming that Etymonline is correct, which is now doubtful, the answer would be: yes, these expressions share the same root, which is Latin ponere, from the Proto-Indo-European root/reflexes *apo-, po, ap-u, pu, "(away) from, off, out of".
However, because to pony up apparently came from the first verse of the Psalm that happened to be sung on a certain pay day, the link in meaning was severed there: in the Psalm, the word pone is used in the sense "provide me with / explain to me the law by which I may lead a good life", i.e. it is an abstract word, not at all connected with paying money in that context. If the first line had been O Domine, viam iustificationem michi pone legem, or something else, the slang term might have been "Yo, dude, o domine me now, or I'll take your wife". I have actually no idea how this slang was used in the 19th century, but you get what I mean.
Even so, it is possible that the use of legem pone to indicate paying money would have died long ago, had not the association been felt between "putting down money" on one hand and ponere, "to put down" in standard Latin, on the other. That would be very hard to prove or disprove; I have no idea.