I got my first BBL treatment 3 weeks ago by a plastic surgeon. How fast do dark spots darken after a BBL treatment?

Yes, some patients experience immediate darkening of these spots while other patients will see darkening over a week following the treatment. 
BBL treatment is a spectrum of treatments. Each visit may concentrate on different problems (browns, broken capillaries/veins, texture) depending on your concerns. The provider will chose appropriate settings to address those concerns.There could be a gradual increase in the strength of the treatment should your skin response level out. Keep in mind that sometimes the effects of the laser may not happen on a surface level visible to the naked eye. I recommend you have your Doctor use a Visia Skin Analysis in order for you to better see all the improvements both on the surface and subsurface of the skin. Make sure you discuss with provider the concerns and goals you have for your treatments so they better address them. I hope this is helpful!