Will my gums grow between my front teeth to fill the triangle after braces?

It is very probable that the gums will not grow to fill in the space between your front teeth. You could have a small area of bonding done to fill in this space. 

That depends on the bone height between two teeth. To analyze that, an IOPA x-ray is a must. so it's my advice to upload an x-ray, so I can guide you better. If the bone is not enough in that area, we can increase gum height with a bone graft or tissue graft.

Often times, in non-growing patients, when there is significant crowding, or overlapping of teeth, there can be a possibility of having a 'Black Triangle' where the teeth taper and go into the gums. There are ways to try and reduce to minimize this, but sometimes we have to accept it.  It is also based on if there was a previous history of gum problems or gum disease, which will make an individual more prone to this. 
The bottom line answer to your question, the gum in an adult will likely not 'grow' to fill the gap. The teeth can be filed so as to bring the roots of the teeth closer together, or the roots tipped slightly towards each other to help.