Athena (and Hera) lost the Judgment of Paris to Aphrodite:
[E.3.2] For one of these reasons Strife threw an apple as a prize of beauty to be contended for by Hera, Athena, and Aphrodite; and Zeus commanded Hermes to lead them to Alexander on Ida in order to be judged by him. And they promised to give Alexander gifts. Hera said that if she were preferred to all women, she would give him the kingdom over all men; and Athena promised victory in war, and Aphrodite the hand of Helen. And he decided in favour of Aphrodite51; and sailed away to Sparta with ships built by Phereclus.
Apollodorus. The Library. Translated by Sir James George Frazer. Loeb Classical Library Volumes 121 & 122. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1921.
If you also count ties, she tied with Poseidon in their contest for patronage over Troezen:
[2.30.6] During his reign, they say, Athena and Poseidon disputed about the land, and after disputing held it in common, as Zeus commanded them to do. For this reason they worship both Athena, whom they name both Polias (Urban) and Sthenias (Strong), and also Poseidon, under the surname of King. And moreover their old coins have as device a trident and a face of Athena.
Pausanias. Description of Greece. Translated by Jones, W. H. S. and Omerod, H. A. Loeb Classical Library Volumes. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1918.