In fact much of the writing of that time deviated off into Latin to formulate their more esoteric points or when quoting a former scholar to buttress their point; VERY annoying as I cannot read Latin and a translation is rarely provided.
Either way it seems highly likely that a Rabbi like Jesus the Christ could, much like a Medieval scholar discussing esoteric point in Latin; speak and discuss theological or philosophical points in Hebrew.
Except that Latin became the non-spoken only in the 20th century. The Croatian parliament, for example, used spoken Latin in the early 19th century. The Ragusa (Dubrovnik) republic on the Adriatic coast was a Latin-speaking community until Napoleon destroyed it in the early 19th century. The Vatican conducted all its business in spoken Latin all the way up to the Vatican II in 1964 and for years to come, etc.
For centuries, Latin was the lingua franca of Western Europe in universities and other institutions, the way English is today a universal language. There simply is no documentation as to the extent of spoken Hebrew, so it is pointless to try to guess to what extent it was spoken, if it was spoken at all. Today, the Vatican still uses all official correspondence in Latin, but hardly anyone in the Catholic Church speaks it.
Again, whether Jesus and Nicodemus spoke Hebrew or Chaldee is irrelevant. The pun that supposedly caused Nicodemus to misunderstand Jesus is possible only in Greek, not in Hebrew or Aramaic. Which raises valid suspicion if that conversation ever took place.