Llorente left Athletic Bilbao for Turin on a free transfer after his contract expired, but initially struggled to nail down a regular starting spot with the Bianconeri.
The Spaniard has since scored twice for the Old Lady and started their last three matches, insisting that he now feels ready to find top form in the Italian first division thanks to Conte's support.
He told beIN Sport: "I'm happy to be playing for this club. It is a great one who believed in me and that was the most important thing.
"Before arriving I regularly spoke with Conte and his belief in me pushed me to sign for Juventus. It was difficult at the start for me to settle in Italy.
"This was a new club, a new language, everything was tough. But I feel much better and I want to demonstrate what I can do."
The 28-year-old is looking to continue his partnership with Carlos Tevez in attack tomorrow night when the Old Lady host Catania at the Juventus Arena.