Nurse, one of my patients has memory loss, she lives alone with no known family. Can I request the opening of a legal protection measure for her?

Yes, but under certain conditions. In principle, only some people can ask the guardianship judge to open a protective measure (art. 430 C. civ.). These are, first and foremost, the person to be protected and his or her immediate family. Third parties should therefore not be able to apply for such a measure.
However, if you are unable to refer the matter directly to the guardianship judge, you may refer it to the public prosecutor who, on the basis of information on your patient's social and financial situation, may forward your request to the judge (art. 431 C. civ.). 

Note: the nature and procedures for gathering this information are specified in articles 1216-1 to 1216-3 of the Code of Civil Procedure.