Yes it is not only possible, but a fact: the name was first known to us as Karayev. Later thay used also Garaev
Nowadays, since Azerbaidjan in an independant country, they start transliteration from there own language and decided the sound of the letter should be transliterated as Q. The letter sounds as a softened K.
In most other languages this sound doesn't exist, so they tried to stay as close to the sound as they thought was correct for English speaking people. The sound actually has an Arabic origin. The language of Azerbaidjan is related to Turkish. More on this matter you can find here:
http://www.mongabay.com/history/azerbaijan/azerbaijan-language_language,_religion,_and_culture.html.
I feel we now should use the Qara Qaraev.
Elroel
Elroel