This question needs to be answered as all electron transfer steps. You have correctly identified the motivation for a reaction to take place. The electrons of nitrogen are not tightly held (compared to O,F,Cl,Br). Boron is planar with an exposed positively charged nucleus. The electrons of the boron are also being pulled away from that nucleus by the fluorine atoms. Therefore it can be anticipated that boron can attract the electrons of nitrogen.
Once the complex has formed, the electrons on nitrogen will be more tightly held due to boron pulling one pair of electrons away from the nitrogen. That will make the remaining electrons on the nitrogen to be more tightly held than before. The boron will now have a completed octet and because the shared number of electrons is greater than the number of nuclear protons, the boron is noted with a negative formal charge. Will boron be able to attract an additional pair of electrons from the nitrogen?
An aside, I don't know this so I am only guessing. In this instance, if BF3 were being added to the amine, I could imagine that another molecule of the excess amine during addition to abstract the proton from the ammonium nitrogen. Then the non-bonded electrons could form a double bond to the boron with an expulsion of fluoride to give F2B=N(CH3)2 + (CH3)2NH2(+) + F(-). This could also be written as a more favorably appearing resonance structure F2BN(CH3)2. I don't think that is what is being expected from this problem, but since a question was asked about whether a proton transfer could take place, I thought I would suggest a rational proton transfer reaction. Again, for this reaction to take place, it is the non-bonded electrons of a nitrogen would be attracted to the proton of the ammonium nitrogen of the F3BNH(CH3)2 complex.