In what language did the conversation between Jesus and Nicodemus happen?

What a great question! You've put your finger on one of the issues that NT scholars face when trying to understand the words of Jesus in the Gospels. On the one hand, almost all evangelical scholars would recognize that we can't be sure when we have the actual words of Jesus, even though we can be confident that we have his 'voice.' That is, in this case, John may have summarized the account in a way that was clear to Greek readers even if the conversation took place in Aramaic. Linguists point out that one can say virtually anything in any language, even though the same number of words may not be used. If that is the case here, John is skillfully showing that there was confusion on Nicodemus' part about what Jesus meant, even though the word-play exists only in Greek. How would the conversation have taken place in Aramaic? I don't know, but it would most likely have been significantly longer and more convoluted, leaving Nicodemus with the question he had.

But there are two other considerations. First, does Jesus really mean only that he must be born from above? That is highly unlikely, since word-plays are John's stock-in-trade and he often, if not usually, means BOTH things. So, Nicodemus could well have understood Jesus partially, but not fully. And John is thus giving hints to the reader that a double meaning is in view, only one of which could have been known to Nicodemus.

Second, it is not at all impossible that the conversation actually took place in Greek. More and more NT scholars are coming to the conclusion that Jesus often taught in Greek. And there is significant evidence that even in Jerusalem--even among the Pharisees, which Nicodemus was--Greek was the only language spoken by them. Thus, we really can't say that this conversation did not occur in Greek. What we can say is that John has accurately, if selectively, portrayed it and that the double meaning he uses was most likely intended to have its full force on the readers. That is, Nicodemus needed to be born again AND born from above.