The local authority will use its shareholding to get the issue discussed at the Scottish Championship side's annual general meeting.
The Queens Trust hopes to secure a director's role or a place as a non-voting member of the board.
The Dumfries common good sub-committee will now back a resolution being taken to the club's AGM.
It does not guarantee a seat on the board for the supporters but does ensure the issue will be debated at the meeting to be held later this year.
The club has said that its door is "always open" for an "open and honest dialogue" with fans.
It has also appointed a supporter liaison officer to improve the relationship between fans and directors.
A fans' bid to get a seat on the board at Queen of the South has been backed by Dumfries and Galloway Council.