The 44-year-old claimed the prize in 1996 when the Old Lady triumphed over Ajax, but he is desperate to repeat the feat at the helm of the Italian champions, for whom he claims continued success is paramount.
"Success is not just important, but the only thing that counts," he told Uefa.com.
"You try to work hard to achieve victories and fill the cabinet with more new and prestigious trophies next to the ones already won.
"Juventus are among those powerhouses who have always needed to win, and along with AC Milan and Inter have won the most in Italy and also on a European level.
"The Champions League has something magical and every player, every coach and every club would like to lift the trophy with the big ears at least once."
Conte also heaped praise on Italy veteran Andrea Pirlo, saying that the 34-year-old boasts talent which is rarely seen in football.
"Players like Andrea are hard to find, because they have such extraordinary ball technique," Conte said.
"He brings calm to the team in possession and anticipates play. Adding those talents to an organised system where everybody knows what they have to do, his qualities come out even better.
"Andrea doesn't run any less than before. I have studied the statistics and he is in the top ten players who have run the most in the Champions League. Physical strength is not his talent, it's his brain and his feet. He makes the collective stand out."
Finally, the former Italy international spoke about the upcoming Champions League clash between his side at Real Madrid next month, admitting he is relishing the prospect of facing old mentor Carlo Ancelotti.
"It's quite a few years since Carlo coached me. I spent two years with him, and they were wonderful, intense times. It was inevitable we'd meet again, coach to coach. That means he is getting old. Me too," he joked.