What is the angle that the earth moves in its orbit during the winter in the radians?

if you assume the orbit is a circle (not quite, but close)

and the seasons are of equal length (not quite, but close)
then winter is 1/4 of the year, meaning the earth moves 1/4 of the way around.
That is, 1/4 of 2π = π/2 radians

If you want to get more precise than that, you have a lot of research to do!