Elvis and Judy became close during filming, but unusually he did not pursue a physical relationship with her. - one of the few times over his film career. This was because he had one unbreakable rule for himself and, indeed, his Memphis Mafia pack – no married women. Although, Elvis' Lamar Fike later claimed the pair "had a thing going."
It seems unlikely because Judy had just married her second husband, Gregory Lafayette, that year and was deeply in love.
Devastatingly, they would both die in a horrific car accident driving across country straight after filming finished on the movie.
Source : https://www.express.co.uk/entertainment/films/1496887/Elvis-Presley-heartbreak-Jailhouse-Rock-costar-Judy-Tyler-die-Elvis-girlfriends197