No. In DB-Z, Goku can only go as high as SSJ3. He learns SSJ4 in DB-GT, but that's all. The rumored DB-AF (AF means April Fools) has been debunked.
There is no such thing as DB-AF and therefore no such thing as SSJ5 except in the minds of those stubborn enough to still believe it exists. There has been almost no new DB material released since the end of GT and most of what was released was recompilations of old stories. Akira Toriyama himself has said that there will be no new DB series released and that GT was the end.
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Dragon Ball Z Kai episode 47
Episode 214: Vegeta's Pride. Goku turns to USSJ (Ultra Super Saiyan) and USSJ2 in the Hyperbolic Time Chamber. He turns to SSJ2 while fighting Majin Vegeta.
Vegeta becomes super sayian 2 when he doesn't kill Majin buu. When he came back to help goku defeat super buu.
Dragonball GT Episode 1: Secret of the Dragon Balls
No Goku does not turn SSJ 10 nor do any other characters, the max is SSJ 4 which only Goku and Vegeta reach.