It's been confirmed in Star Wars canon that Qui-Gon Jinn would never have left the Jedi Order to join Count Dooku. It's a rare thing for a Jedi to leave the Order, and until Attack of the Clones only 20 Jedi had ever done so. According to librarian Jocasta Nu, Dooku's decision to return to his homeworld Serenno was a painful moment for the Jedi, not least because he was one of Master Yoda's own students.
In Attack of the Clones, Dooku outright claims that, had Qui-Gon Jinn not been killed in battle by Darth Maul in Star Wars Episode I: The Phantom Menace: when Obi-Wan is captured on Geonosis, he states "I wish he were still alive. I could use his help right now. [...] He knew all about the corruption in the Senate but he would never have gone along with it if he had learned the truth as I have." That truth is that Darth Sidious controls the Republic government.
Subsequently, over the years there's been a lot of discussion in Star Wars fan communities over whether or not Dooku would have corrupted his former Padawan. He's definitely not the most rule-abiding Jedi; in The Phantom Menace, Obi-Wan Kenobi cautioned Qui-Gon that he would have been sitting on the Jedi Council if he'd just follow the Jedi Code. Qui-Gon trusted in his own instincts, believing he had to act in accordance with the will of the Force in any situation. To Qui-Gon, rules were reinterpreted as guidelines, and the structures of the Jedi could act as obstacles to serving the Force.
The bond between Master and Apprentice is a powerful one, and in theory, it is possible that Count Dooku could have reached out to Qui-Gon Jinn as he did to other Jedi (had he survived). But Claudia Gray's latest novel, Master and Apprentice, has confirmed that Qui-Gon would have rejected Dooku outright. The fundamental issue is that Qui-Gon believed every being had to choose whether to serve the light side or the dark, and that he had resolved to follow the light. As he told his old friend Rael Aveross in the book, "I don't turn toward the light because it means someday I'll 'win' some sort of cosmic game. I turn toward it because it is the light."
Dooku would have known full well that Qui-Gon was dedicated to the light side of the Force. Master and Apprentice contains a number of flashback scenes back to when Qui-Gon was apprenticed to Dooku. In one, the 14-year-old Qui-Gon wound up captured by a notorious and brutal bounty hunter named Shenda Mol. She intended to torture and then kill the young Qui-Gon, but Dooku intervened. Enraged because his apprentice had been threatened, Dooku succumbed briefly to his own inner darkness, torturing Shenda Mol - and the description seems to suggest he even used Force Lightning. Although Qui-Gon didn't see it, he was absolutely horrified at the sounds and the smell, and the relationship between the two was never the same.
Dooku did approach one of his old apprentices, though. Rael Aveross had been Dooku's Padawan before Qui-Gon, and was in a very vulnerable place after the death of his own apprentice. He'd never been quite so wary of the dark side; in fact, when Qui-Gon told Rael about Dooku's torturing Shenda Mol, he wasn't at all concerned. Fortunately, Qui-Gon had a strong influence on Rael, and his words inspired Rael to reject Dooku's offer. He remained loyal enough to his old Master not to breathe a word of what he knew about Dooku, though, even to Qui-Gon. Had he only opened up, the Jedi would have confronted Dooku seven years before the events of Star Wars: The Phantom Menace, and galactic history would have taken a different path.
