I was wondering if the names of the seasons should be capitalized.

The names of seasons are not usually capitalized.
I love autumn leaf colors.
Today is the winter solstice.
In the springtime the crocus are the first flowers to appear.

They would be capitalized when personified.
Then Spring—with her warm showers—arrived.
Old Man Winter

They would be capitalized in a title.
All City Elementary Autumn Open House