Gregor Murray, the deputy convener of education on Dundee City Council, attacked the idea, calling it "the biggest load of p*** I have ever seen".
Mr Murray previously worked for Shona Robison, the Cabinet Secretary promoting the quota.
At the SNP's recent spring conference, the First Minister said he wanted 40% of the places on company boards to be held by women, and announced he was leading by example by adding two female ministers to his cabinet.
Ms Robison, who was one of those promoted, now holds the equalities portfolio at cabinet.
The move was widely interpreted as an appeal to female voters ahead of the independence referendum, and attacked by the opposition as cynical.
Ms Robison last month launched "Women on Board", a consultation on a 40% quota for public boards such as NHS bodies and quangos. Ministers have also asked for views on whether future legislation for a 40% mandatory female membership on public bodies should be extended to the private sector.
Although females make up 52% of the population, only 36% of public board members and just 21% of board chairs in Scotland are women.
Expressing her determination to achieve gender equality, Ms Robison wrote: "We need to ensure that we do not have to wait years to achieve equality on our boards ... By taking action on this issue we will be improving economic participation by removing barriers that stand in the way of women realising their potential."
Mr Murray, 27, was scathing. "Well, this has to be the biggest load of p*** I have ever seen in my life," he wrote on Facebook next to a link to the consultation.
A few minutes later, in a reference to Ms Robison, he added: Sorry Shona pal, I love you very dearly, but that's just f***ing stupid."
Mr Murray also attacked the idea on Twitter, writing: "Sorry @ShonaRobison but this has to be most stupid suggestion I've seen in a long time. #NotEquality #Tokenism".
Mr Murray, one of two SNP councillors in Dundee's North East ward, was elected in 2012.
He was previously a researcher for Ms Robison, MSP for Dundee East, and for Dundee West SNP MSP Joe Fitzpatrick, the Government's chief whip.
His council role means he sits on the 42%-female board of the Dundee Science Centre.
Labour equalities spokeswoman Jackie Baillie said: "When even senior nationalists are admitting that Alex Salmond's recent conversion to gender equality in his cabinet is tokenism, you know that he is fooling no-one and most women will see the SNP's clumsy attempts to woo them for what they are."
Speaking about his comments, Mr Murray said: "While I have my own views on the subject, I believe that I should have expressed them in a more responsible way."
An SNP spokesman said: "Women now make up 40% of the Scottish Government Cabinet, and ensuring that women make up at least 40% of the boards of our public bodies will help to drive that change forward - something that the SNP takes extremely seriously, even if Labour clearly does not."