Is this a legitimate Apple support number?

We are end users like you can cannot know all Apple's numbers. The best way is to simply to a web search for the number. I got this:

Spam Reports

Although Apple has operations in the Austin TX area code, the search results make me suspicious.

How did you get the number? If a robo-call or a call where your did not initiate contact, it is usually bogus. Apple will not call you unless you call them first.