No, joda time does not provide this functionality. It provides you info about time which has to do with how much the earth is rotated on its axis. Seasons have to do with the earth's rotation relative to the sun. So even though "Los Angeles" is in the northern hemisphere, Joda time doesn't know that. It only knows that its time zone is UTC-08:00
which exists in both hemispheres.
Luckily the earth spins (reasonably) predictably and cyclically around the sun. So at any time you can determine the season for a given time and Hemisphere. In the referenced stack overflow they basically fixed the hemisphere so that the only two variables where time and season.
If you want to take into consideration the hemisphere you can use the existing solution and at the end write a function that will invert the season if your location is in the southern hemisphere instead of the northern hemisphere.
Solution left to the reader as an exercise