How do I integrate the Color module in my theme?
There's actually a video on YouTube about how to do this in Bootstrap.  There's also a more general video for other themes by the same person.

You'll need to use a LESS or SASS subtheme; the most basic subtheme that just uses regular CSS will not work because the Color module will change the values of the variables.

You will also need a way to compile the LESS/SASS on the server that you use the Color module on, so if you plan to make adjustments on your production server, you will need something like the LESS module, which does not yet have an official release for Drupal 8 (might still work, needs testing).

You can also find a feature request on drupal.org for putting this directly into Bootstrap, but it hasn't made any progress in the past couple years.