What accent is Kanbaru referring to, and where did she live prior to her parents' deaths?
Could not recall any mention in the anime, either. A quick look at the light novels, however, seem to reveal some information, albeit also vague. To quote from Chapter 7 of Bakemonogatari: Monster Tale Part 02: 


  The way I spoke was different. I might talk like this now, but when I was still with my parents, we were all the way out on the tip of Kyushu, probably to get as far away as possible from this home. They talk in a thick accent there, and well...l wouldn't call it bullying, but I was made fun of, and I didn't have any friends.


Assuming that the tip she's referring to is the southernmost part of Kyushu, 'all the way out on the tip of Kyushu' may refer to a place somewhere on the Okinawa prefecture (possibly within the Yaeyama islands). So, it could refer to an accent unique to people who speak the Okinawan language or dialect.