Yes, but it's debatable exactly how much. Some sources say it was huge, others say it was just a little bit.

According to the book Operation Hollywood: How the Pentagon Shapes and Censors the Movies, "..after the film's release, the US Navy stated that the number of young men who joined wanting to be Naval Aviators went up by 500 percent."

LA Times ran an article in 1987 that also confirmed a bump, but not as drastic of one:

recruiters in some cities manned tables outside movie houses during "Top Gun" premieres to answer questions from would-be flyboys emerging with a new-found need for speed from an F-14 warplane.

"There seems to have been a big rush in those categories that I have to attribute to the movie. I've asked several of these individuals if they've seen the movie and if that's why they came down to talk to us again and they've said 'yes'.

..

"They couldn't specifically say it was a direct result of 'Top Gun,' but they suspect it probably had a lot to do with it because when they would talk to applicants, about 90 percent said they had seen the movie," Marlowe said.