Thanks for the question, and welcome to Apple Support Communities.
The following article addresses your symptoms:
When opening FaceTime from the Dock, the following alert message may appear:
"An internal FaceTime error occurred.
There was a problem with FaceTime. You need to quit and open FaceTime again."
Click the FaceTime icon in the Dock one time, then wait for the application to finish opening.
This can happen if you attempt to open FaceTime a second time while it is already loading, such as by clicking the FaceTime Dock icon more than once.
If you continue to have issues, it may be a good idea to remove FaceTime and reinstall it from the Mac App Store