I think it's impossible to answer your question without knowing: (1) where and when your prof went to school; and (2) what you mean by a degree in "classics."

I can say that at my school a Latin major and a Greek major were both language majors, requiring 21 hours of upper level Latin or Greek, plus 9 hours of related culture/history/lit.

There were separate majors in Classical Civilization, but I think languages were optional in these courses.

I'm going to talk from a UK standpoint, so if you're doing s course in the US I likely can't lend much insight.

A lot of courses in Classics/Ancient History will cover the Byzantine Empire, although many will only briefly touch on it. Most Classics and AH courses focus on Ancient Greece, the Roman Republic and the Roman Empire, and the collapse of the Western Empire is where a lot of them stop. Therefore, you may not be able to find a great course on the Byzantines until postgraduate. At least, not an in-depth one.

In terms of Latin and Greek, most universities in the UK that do Classics or AH will allow you to take modules in both Greek and Latin all the way up until your final year, and I believe that they are compulsory for Classics in some places. For example, at Newcastle University you choose three extra modules to do alongside the core modules in the Ancient History course, and you can select Latin and/or Greek for them.