Is ran quickly an example of a simple predicate?

No. It might be a complete predicate in "The boy ran quickly."

The simple predicate is only the verb and any helping verbs it uses, not adverbs.

Examples:

The cow has been milked today. (simple predicate: has been milked)

The man is walking to the store. (simple predicate: is walking).

The man enjoys jogging. (simple predicate: enjoys; jogging is a gerund/noun)