The council said it had hired 70 new workers to help with the new rounds.
A web app will also be launched to give people reminders and updates on the collections.
The recycling plan will be introduced at flats and communal properties on a phased basis over the next year, the council added.
Small food caddies will be made available for those who need one and the council will also increase the number of places where people can collect free replacement liners.
Quinn said: “By placing food waste in the brown bin, it can be recycled into environmentally friendly local compost.
“This approach is not only better for the planet, but it also saves council tax payers money, so we’re fully embracing the opportunity to expand food waste collections for our residents.”
Waste from the blue recycling and black general bins will continue to be collected every two weeks.
Listen to highlights from Hull and East Yorkshire on BBC Sounds, watch the latest episode of Look North or tell us about a story you think we should be covering here.