Played by the late Christopher Lee, Count Dooku – the former master of Qui-Gon Jinn – is one of Star Wars' most intriguing characters given his fall to the dark side. Dooku is one of the most dangerous villains of the prequel trilogy, his backstory finally explored in Cavan Scott's audiobook Dooku: Jedi Lost and the Tales of the Jedi animated series. Like Qui-Gon, Dooku came to realize the Jedi Order was flawed; unlike his apprentice, Dooku chose to embrace the way of the Sith while Qui-Gon simply became a maverick.
Oddly enough, Dooku's motives remain somewhat enigmatic. Star Wars can't decide whether Dooku knew about Order 66; some portrayals seem to show him hoping to reform the Jedi, while in others he's a willing accomplice in Palpatine's plan to destroy them. This is perhaps partly because, as seen in Star Wars: Episode II – Attack of the Clones, Dooku was willing to even tell a Jedi Master that a Sith ruled the Republic. It's unclear whether he was helping Palpatine hide in plain sight, or whether he was trying to manipulate events to his own end, using Palpatine to advance his own caseu.
Dooku Really Does Seem To Have Believed Qui-Gon Would Join Him 
What is striking, though, is that Dooku genuinely seems to have believed Qui-Gon would have joined him had he lived. Speaking to Obi-Wan Kenobi in Star Wars: Episode II – Attack of the Clones, he attempted to open Obi-Wan's eyes to the failings of the Jedi Order. In his speech, he genuinely showed grief for the loss of Qui-Gon Jinn, Obi-Wan's old master. Dooku seemed very confident Qui-Gon would have left the Order and joined the Separatists after learning the same "truth" he had.
While it’s possible Dooku said this to shake Obi-Wan’s convictions – hoping he too would join the Separatists – there is a grain of truth in his argument. Like Dooku, Qui-Gon knew the Senate was corrupt, slow, and had often failed the people of the Republic. Qui-Gon had pieced much of the puzzle together by studying the ancient Jedi prophecies, perhaps even suspecting the continued existence of the Sith. Dooku presumably believed the information about the Sith controlling the Senate would have been the deciding point. Claudia Gray’s novel Master & Apprentice even showed Dooku attempting to reach out to his apprentice multiple times before Qui-Gon's untimely demise, and he may well have planned to make this argument.
Dooku Was Deluding Himself Over Qui-Gon 
While Obi-Wan and the Council were skeptical of Dooku’s claim about the Senate being under the control of the Sith, Qui-Gon would have quickly understood this point – and would likely have even believed Dooku. However, Dooku was ultimately deluding himself when he believed his former apprentice would side with him. Qui-Gon held too firmly to the ancient Jedi prophesies to consider abandoning the Order or joining the dark side. If anything, Qui-Gon would have continued trying to reform the Order from within.
