The ambitious cinch League One outfit have won just one of their last nine matches in all competitions and have slipped to fourth in the table, 13 points adrift of leaders Cove Rangers.
Wednesday's 1-0 defeat away to promotion rivals Airdrie has proved to be the last game in charge for Ellis, who replaced Ray McKinnon as manager in May following their promotion from League Two.
Murray joined Ellis as assistant in the summer after a three-month stint as Gary Naysmith’s number two at Edinburgh City. He was a youth coach at Hibs for six years before that.
A statement on the Spiders' website said: "The club is grateful to Laurie for his hard work and commitment but has reluctantly taken the decision to appoint a new management team to take the club forward."
Peter Grant Jnr and Gardner Spiers will take interim charge of the team until a new appointment is made.
Chief executive Dempster, who joined the club after her departure from Easter Road, said: "We are grateful to Laurie and Grant for their work, we appreciate their efforts and wish both every success in future.
"We are now looking forward to bringing in a new management team that will help us build the next successful chapter as we seek to realise our ambitions for Queen's Park on and off the pitch."
Queen's recently appointed former AZ Alkmaar sport development director Marijn Beuker as their sporting director.