Our former loving brothers and sisters have been instructed to think that, Jesus likely spoke a form of Hebrew and a form of Aramaic. (Aid to Bible Understanding-103-105)
Of course, there is no way to demonstrate the truth of that assertion or any other assertion about the languages that the common people of the land spoke. But, historically we can demonstrate what languages are more likely to have been used.
We do know that in 332 BCE, a Greek army commanded by Alexander the Great marched through Palestine (after beseiging Tyre), beginning centuries of Greek influence and political control. By the time of Jesus political control had been ceded to Rome and soldiers speaking Latin controlled the a land, but Greek culture and language seemed to have remained prevalent. This is demonstrated when, a few centuries later the Eastern Roman Empire, with its capital located at Constantinople, used Greek as its language.
So what about the claim in the Aid book? Ex-witnesses may be interested in the arguments of a scholar G.Scott Gleaves (Dean and Associate Professor, Kearley Graduate School of Theology, Faulkner) in this essay:
Did Jesus Speak Greek?
http://www.bibleinterp.com/articles/2015/09/gle398009.shtml
He argues - Contrary to contemporary scholarship, I find that Greek was more widely used in both written and oral form by Jesus, his disciples, and the Jews who inhabited first-century Palestine. Interestingly, the evidence reveals that Greek became the dominant language spoken among Jews and Gentiles in Galilee in the first century CE.