Possible, of cause but it'd be heavy effort and a step into the wrong (in my opinion) direction. But you asked whether it's possible...
So... You'd need to implement an appropriate to one of the available (redis, mongodb, leveldb) interfaces: https://github.com/NodeBB/NodeBB/tree/master/src/database I guess with sth. like https://github.com/mysql/mysql-js it shouldn't be too hard to do so. But still much work to do (and to maintain) and I doubt that any of the devs around here is willing to do this (at least without :gift:).
@babaggeii said:
It would surely open nodebb up to a much larger market.
I doubt that it'd get much larger since those who insist in SQL-database types are mostly those who are afraid of new (and better in many cases) technology such as node.js.