How old are you?
I'm not being a dick, but it is relevant and since you brought up the yellow belt thing it makes me think you guys are pretty young.
The reason this makes a difference is because there is a world of disparity between a 30 year old person training with their spouse and a 15 year old training with their crush.
Does this girl train at your academy?
Common interests are great for relationships.
It will give you guys something to talk about and do with each other.
However, if things turn south, you don't want it to interfere with either persons training.
Early teens breakups can be kind of brutal, and I would hate for either one of you to quit training BJJ because they don't want to run into the other at class or a tournament. The difference between a married couple is, the fact that most will get over their arguments and end up training the next day together. Younger people seem to react more irrationally and in permanent manners.
"He's such a jerk, I never want to see him again and I am burning my gi." type thing.
But in the end, these are problems you will face with any relationship until you find your mate.
So take the advice for what its worth, but I would keep separate training agendas for the most part. Let her train at her gym (if it is a different gym than yours) and you stay at yours.
Who knows, you guys could end up being soul mates, but I always try to hope for the best, but be prepared for the worst.