I was looking up the name Barbara and Barbarian and see that there is a connection,
SEE: http://www.alphadictionary.com/goodword/word/barbarian
but what about the word “barber?”

Nope, no relation. Barber is ultimately from the Latin barba, meaning beard.

Thanks for such a quick reply.  BTW, I just now won a debate between my husband and myself.  ;)