This was announced by the organisation on Monday.
Her successor will be the US nun Roxanne Schares, previously deputy executive director.
Murray is a member of the Loreto Sisters and worked for many years as a teacher, headmistress and university lecturer. She took part in the Synod on Synodality initiated by Pope Francis on greater participation of all believers in the Catholic Church.
Murray is also an advisor to the Vatican's education dicastery. Since 2014, the nun has held the office of UISG Secretary General.
"Expertise and sensitivity"
UISG President Oonah O'Shea thanked Murray for her "generous and passionate" service.
With her competence, professionalism and ecclesial sensitivity, Murray had promoted processes of communion and cooperation between the congregations and inspired a new spirit of synodality.
O'Shea also praised Murray's commitment to ongoing dialogue with the various realities of the global church and with international institutions.
"Under her leadership, the UISG has strengthened its commitment in important areas such as the integrity of creation, justice and peace, the protection of the most vulnerable and the formation of new generations of leaders in consecrated life," said O'Shea, who has led the association since mid-May.
More than 1,900 communities from 97 countries belong to the UISG.
In total, there are more than 500,000 Catholic women religious worldwide.