Maybe the question to ask isn't whether or not it's something that is specifically motivating him, because there's obviously multiple motivations he or anybody has in doing anything, but how aware is he of the record?
A) He obviously knows that this would be the 4th consecutive Super Bowl should they make it, assuming he can count on his fingers and remember the last few years.
B) He obviously knows the Buffalo Bills and Jim Kelly and that the early 90s teams remain the franchise's greatest accomplishment, unless we're counting the 1960s, or OJ Simpson's 2000 yard year... he's played football in Buffalo for 20 years, he certainly knows our lore.
So how much has he put these thoughts together? How often is he thinking about the Bills at the same time he's thinking about Super Bowl records? Has he smirked his little goblin grin at the idea of climbing to another plateau, one that would count as three wins at once — further solidifying his GOAT status, and further torturing Buffalo? Does he view Bills fans with derision, apathy, contempt, pity?
I don't know but I would venture to say that Bills fans think about Brady more than he thinks about Bills fans.