On one of these journeys I was going to Damascus with the authority and commission of the chief priests. About noon, King Agrippa, as I was on the road, I saw a light from heaven, brighter than the sun, blazing around me and my companions. We all fell to the ground, and I heard a voice saying to me in Aramaic, ‘Saul, Saul, why do you persecute me? It is hard for you to kick against the goads. NIV
'Kicking against the goads' does seem to derive ultimately from Greek drama, but it seems to have been a fairly common expression, so it may well not be a quotation. Both Euripides and Aeschylus have it. One reason for thinking that it might actually be a quotation rather than a general proverb is that the sense of it as we find it in Euripides' Bacchae fits Paul's situation surprisingly well, and it would make sense for Paul to add the comment as an explanation to Festus what was going on:
I would sacrifice to the god rather than kick against his spurs in anger, a mortal against a god.
Pentheus is trying to stamp out the worship of Dionysus; Dionysus himself has come, in disguise, to spread the religion. Pentheus put him in prison, but, of course, being a god he escaped. Pentheus threatens him with punishment again, and then Dionysus makes the reply above. One can well imagine Paul using the quoted phrase to allude to something like this in order to explain his point to Festus -- or, perhaps adapting the language of Euripides in order to make the point more clear in Greek.
The similarity with Aeschylus is less dramatic. From the Agamemnon:
You speak like that, you who sit at the lower oar when those upon the higher bench control the ship? Old as you are, you shall learn how bitter it is at your age to be schooled when prudence is the lesson set before you. Bonds and the pangs of hunger are far the best doctors of the spirit when it comes to instructing the old. Do you have eyes and lack understanding? Do not kick against the goads lest you strike to your own hurt.
We still have here the sense of the absurdity of rebellion; it's just not as close a fit as Euripides' use of it in the context of overwhelming divine power, and the notion that the struggle is with something well beyond the merely human.
But, of course, we have to keep in mind that it passed into a proverb, and therefore is perhaps only connected historically, not intentionally.