Judy Lindsay says Ms Williams hosted an online meeting with her and two other families who had lost family members in accidents caused by drink-drivers hours before the mayor crashed her own car into a tree east of Brisbane on Thursday night.
Ms Williams says she'd had "several glasses of wine" at a party and admits she made a mistake getting behind the wheel, but refuses to resign, saying it would be a "rash decision".
Police are investigating the incident and no charges have been laid.
Ms Lindsay, whose daughter Hayley was killed in a crash involving a drink-driver in 2009, says the mayor rang her in tears on Friday night to apologise.
"I just said I can't have any contact with you any more. You can't be doing the job you're doing. I trusted you and this is what you've done," Ms Lindsay told Nine's Today program on Monday.
"She's got to go, she's got to go, she has no respect for anybody in this community by doing what she's done.Â
"And if, you know, having a meeting with us, a Zoom meeting with us, wouldn't that stop you from getting in the car drunk and driving half an hour after having a meeting with us?"
Federal Opposition Leader Petter Dutton and Nationals leader David Littleproud have called for Ms Williams to resign, along with state Environment Minister Meaghan Scanlon and fellow Labor MP Don Brown.
Mr Littleproud said her position was untenable, given her advocacy for victims of drink-drivers.
"If you want to sit there and talk with victims who have paid the ultimate price, the ultimate price of losing their loved ones ... you've got to live up to your responsibility when you're in public office," he told Today.
"We know you're in public office, we know you don't always get it right, and I don't always get it right, but this is a big miscalculation ... I think her position is untenable.
"I would hope she would do that under her own volition."
Mr Peter Dutton said drink-driving was "completely unacceptable" and Ms Williams needed to set a strong example "in terms of actions, accountabilities and consequences".
"My message, particularly as a former police officer, is to not have that extra drink. Don't drink and drive. I mean, there's no excuse for it, and the consequences can be deadly," he told reporters on Sunday.