Jeev, struggling to recapture his vintage form, that once took him to No. 28 in the world, led in the initial stages of the match and was even 2-up. Quesne, however, not only caught up with him but also went ahead and finished the match with four holes to spare, winning 4&3.
The other Indian in the fray, Shiv Kapur was beaten in the first round by Richie Ramsay.
Jeev won the second and fourth holes to go 2-up but he did not win a single hole after that, while Quesne won fifth, sixth, eighth, 11th, 12th and 15th to wrap up the match.
Doak will face David Howell in round three after the English took down Scotland's David Drysdale 6&4. Doak is joined in round three by countrymen Richie Ramsay and Marc Warren after their respective wins over Edoardo Molinari and Fabrizio Zannoti.
Warren will face Nicolas Colsaerts, while Ramsay takes on Morten Orum Madsen. There will an all-Swedish last 16 match when Johan Carlsson (conqueror of Number One seed James Morrison in round two) takes on former European Number One Robert Karlsson.