1) The sun had nearly set when we reached home. Adverb qualify adverb

2) Modern trains can travel incredibly fast. - adverbs qualify adjective
3) The judges were utterly amazed at the performance.- Adverb qualify adjective
4) The bitterly cold winds have killed all the flowers.- Adverb qualify adjective

Are the above adverbs of degrees are correct?

Set

Yes, the adverbs of degree in the sentences are used correctly.

Yes, the above adverbs are correct examples of adverbs of degree. Here's how you can identify them:

1) The sun had nearly set when we reached home.
The adverb "nearly" is qualifying the verb "set," indicating the degree to which the sun had set.

2) Modern trains can travel incredibly fast.
The adverb "incredibly" is qualifying the adjective "fast," indicating the degree or intensity of the fastness.

3) The judges were utterly amazed at the performance.
The adverb "utterly" is qualifying the adjective "amazed," indicating the extreme degree of their amazement.

4) The bitterly cold winds have killed all the flowers.
The adverb "bitterly" is qualifying the adjective "cold," expressing the degree to which the winds were cold.

See my answer below.