I undertand where you are coming from, but short of searching every single Church Register in Ireland she does not seem to be found on available sources.
Not sure how reliable the references to Cork and the Army are
https://paperspast.natlib.govt.nz/newspapers/EP19260907.2.136?query=eileen%20gertrude%20murray&phrase=2&snippet=trueFrom the Evening Post 7 Sep 1926Death occurred on Saturday morning at Christchurch of Mrs Eileen Gertrude Murray , widow of the late Mr Henry Murray , formerly a well known citizen of Christchurch ....The daughter of an Army Officer , Mrs Murray was born in County Cork , Ireland , and came at an early age with her family to Australia. As a young woman she came to New Zealand ... the article goes on to mention her children , Stella Murray , celebrated contralto , Mr H St A Murray ( an athlete I believe , ) and Mr Bernard Murray.Not sure how reliable the references to Cork and the Army are