Francesca,
If she had his child there is a good chance in that era that it was put up for adoption. Your Marine might be able to find that out from
Adoption and Family Records Services (AFRS)
Department of Human Services
20/570 Bourke Street
Melbourne 3000 Australia
Telephone: (03) 8608 5700
http://www.cyf.vic.gov.au/placement-support/family-records/birth_parents
I think the international dial for that local phone number is 613 8608 5700.
I don’t know if that source would lead him to Rose’s present whereabouts.
He could also try the Births, Deaths and Marriages Registry: https://online.justice.vic.gov.au/CA2574F700805DE7/HomePage?ReadForm&1=Home~&2=~&3=~
He might have problems in getting information because of privacy and fraud considerations which restrict access to certificates: https://online.justice.vic.gov.au/CA2574F700805DE7/page/About+Us-Registry+policies-Access+policy?OpenDocument&1=70-About+Us~&2=40-Registry+policies~&3=10-Access+policy~
Probate records may be more readily accessible, although he might need to employ a professional searcher to search the records. This will help only if she died without marrying (if she married the record will be in her married name) and left a will and left assets in Victoria, or if she died without a will and letters of administration were granted to someone to administer her estate. http://www.supremecourt.vic.gov.au/wps/wcm/connect/Supreme+Court/Home/Practice+and+Procedure/Probate+Office/SUPREME+-+Probate+Office+FAQs
The offices I’ve mentioned are for the State of Victoria, of which Melbourne is the capital. There are corresponding offices for births, deaths, marriages, and probate in the other states and territories, which would be the ones holding the record if she was born, married or died there, but I don’t know about their adoption information records.
Your Marine might be a lot better off using a professional tracing service which can do the work for him and which has better resources than just the public offices I’ve mentioned, such as the Salvation Army service http://salvos.org.au/familytracing/about-us/family-tracing-service/ or the Red Cross http://www.redcross.org.au/ourservices_aroundtheworld_tracingrefugeeservices_nmpw.htm