I wasn't born here, but that was due to military service so it shouldn't really count. My father's father was born here, so definitely 3rd but possibly 4th. I do not know if his mother was born here (no one knows much about her as she died young). His father was not, he was born on the ship on the way over from Armenia. My kids were born here so they are 4th.
More importantly, my entire NYC-line has always lived on Staten Island. We can bemoan the Brooklyn invasion proudly.