Her great-grandfather William Gowan was a sculptor, and her grandmother Frances Gowan was a painter. She was educated at the Advanced School for Girls (which had a strong drawing strand), and studied art with the painter Margaret Preston. She began exhibiting with the South Australian Society of Arts as early as 1901; in this period, her work clearly showed Preston's influence.

In 1904, after her mother's death, she went to Europe to study art in company with Preston. They spent the first few months in Munich, where Davidson studied briefly at the Künstlerinner Verein, before moving on to Paris. There she studied at the Académie de la Grande Chaumière, under René-Xavier Prinet, where she met and began a lifelong friendship with Philippe Besnard's future wife, Germaine Desgranges. She also took classes with Raphael Collin, Richard Miller, and Gustave Courtois. A year after her arrival in France, she was exhibiting at the Salon de la Société des Artistes Français, and the year after that at the Société Nationale des Beaux-Arts. In 1922 she would become the first Australian woman elected a member of the Société Nationale des Beaux-Arts, and she would later serve as its secretary. She was a founding member of the Salon des Tuileries, at which she would exhibit almost every year between 1923 and 1951.