This would depend a bit on your background. If you are familiar with fuzzy sets, then Fuzzy Graph Theory by Mathew, Mordeson, and Malik offers a good overview. They also have a follow-up volume with more specialized applications.

If you are not familiar with fuzzy sets, you could start with Introduction to Fuzzy Graph Theory, just to get a feeling for the field, but then you would want to first study fuzzy sets, perhaps by following this online course. (I have not found an online course specifically on fuzzy graphs.)