In 2375, when Vic Fontaine met Nog for the first time, he discovered that Nog was already quite familiar with Vic's work. It was later revealed that Nog was introduced to Vic's work from the recordings Vic made for Julian Bashir to play for the Starfleet troops during their supply run, and subsequent siege of AR-558. Based on this exposure, Vic swore that if Julian was his publicist, he'd be "bigger than Elvis." (DS9: "It's Only a Paper Moon")
In real life, Presley was a big Star Trek fan, even naming a horse "Star Trek". 
Several of Star Trek's alumni worked with Elvis over the years, including Laurel Goodwin, who co-starred in the film, Girls! Girls! Girls! and James B. Sikking, who appeared in the film, Charro!. Lance LeGault was a close friend of Elvis', as well as his stand-in and stunt double. LeGault can also be seen playing tambourine for Elvis on the 68 Comeback Special. TOS director Gene Nelson helmed two of Elvis' films, Kissin' Cousins, which co-starred Yvonne Craig and Teri Garr, and Harum Scarum, which featured Michael Ansara and Theo Marcuse. Craig, who was also Presley's girlfriend at one point, also co-starred with him in It Happened at the World's Fair. William Campbell also acted alongside Elvis in his movie debut, Love Me Tender. TOS actress Maurishka also appeared alongside him in 1968's Stay Away, Joe.