What kind of fescue do you have? Is there a lot of sun or is it shaded?

Most fescues don't spread, so over time, the KBG will tend to dominate, but depending on the kind of fescue, you could end up with a lawn with grasses that grow at different rates and look very different.

I believe that this is one of the red fescues. The area receives about seven hours of sun throughout the growing season. Thanks for your response.

I think 7 hours of sun a day should be enough for KBG. It may or may not crowd out the fescue. If you want to replace the fescue, you'd be better off killing it and reseeding, but if you do something like core aerate the area and overseed with KBG, then maintain it with inputs that favor the KBG, it will probably dominate over time. For example, if you feed it with about 4 lbs of N per 1000 sq ft per year, the KBG will thrive and the red fescue will struggle.