If Derry win, under new boss Rory Gallagher, the Oakleafers will take on the winners facing Donegal or Tyrone in the semi-finals.
Monaghan face 2019 Ulster finalists Cavan in the preliminary round with the winner facing Antrim in the quarter-final.
Down will travel to Enniskillen to take on Fermanagh.
Donegal against Tyrone is the rivalry that has dominated the last decade and the 2020 quarter-final showdown in Ballybofey will be their fifth time in six years to meet in the Ulster SFC, and the eighth time since 2011.
They also met in the Super 8s in 2018, one of thee occasions when Tyrone emerged victorious, but Donegal have held the upper hand winning five of their eight Ulster championship battles – including this year’s one-sided semi-final in Breffni Park.
The Ulster Council decided that the teams drawn in this year’s preliminary round – Derry and Tyrone – would be excluded from it in 2020.
Monaghan, who will have Seamus McEnaney back in charge for the second time, will host neighbours Cavan, who reached the provincial decider this summer for the first time since 2001.
In a repeat of 2019, Antrim await the winner of the preliminary round tie and are entitled to a home draw.
With the Casement Park redevelopment project still stalled, they elected to play their ‘home’ game against Tyrone in the Athletic Grounds in Armagh.
Rory Gallagher’s Fermanagh team knocked Armagh out of the Ulster championship in 2018, he will try to repeat the trick with the Derry footballers in another quarter-final tie.
Gallagher’s former assistant Ryan McMenamin has taken on the manager’s role and his first championship game in charge will be at home against Down.