Does "unnerve" mean "to make nervous"? If not, what are some verbs for "to make nervous"?

Depending on what you are trying to convey, ruffle could be an option. It means

to annoy or upset someone, or to make someone very nervous:

  • He's easily ruffled by criticism. (Cambridge)

As for unnerve, it is more similar in meaning with the verb intimidate. Cambridge defines it as

to make someone feel less confident and slightly frightened