Is it true that in NIT Trichy, Calicut, and other south colleges the professors teach in Malayalam or other regional languages?

Yes, there are many Malayali teachers at NIT Calicut and they sometimes use a few Malayalam words while teaching, but this isn't a big thing to worry about. There is wide variation in faculty and it is possible for them to use some of their regional words by accident, but teaching all the subjects in any other regional words is very unlikely to happen. 

English is the most spoken language by them and all the classes are also conducted in the same. Some teachers might not be as fluent as you would expect them to be, but they are not very bad at it and easily get their job done. Some teachers speak in Hindi also and sometimes use it to explain complex topics to the students. 

There is no matter of concern as they do speak English very well except for a few, but there is a Malayalam accent in their language.