"Contesting the palm" -- looking for a definition and possible origin of this archaic phrase

It appears to refer to palm meaning:

• a symbol of triumph or superiority; also : victory, triumph (M-W)

Ngram shows a number of "contest the palm" usages especially in the 19th century. I think is it is an expression which is just to be taken literally.

Ngram the expression "the palm of victory" was popular during the 19th century.

• The Lucioda of Mrs. Mountain cannot be surpassed, and Mrs. Bland, in Madge, may be allowed to contest the palm of victory with Mrs. Martyr, who has deservedly acquired great reputation in that character. The Bravura, by Giordani, .. The Monthly Mirror -1802

As for its origin, a palm branch as a symbol of victory dates back to the the earlest civilisations:

• The palm branch is a symbol of victory, triumph, peace and eternal life originating in the ancient Near East and Mediterranean world. The palm (Phoenix) was sacred in Mesopotamian religions, and in ancient Egypt represented immortality. In Judaism, a closed frond of the date palm is part of the festival of Sukkot.
• A palm branch was awarded to victorious athletes in ancient Greece, and a palm frond or the tree itself is one of the most common attributes of Victory personified in ancient Rome.

(Wikipedia)

Palm (second entry, meaning 3: see slso 2 and 4); merit or victory. Another way of saying it would be 'to vie for the gold medal'.