Chickens are not like mammals wherein the females or males have to come into "season."
If you have a young, healthy male, he will want to mate any time of day or season.
Your hen likewise, if young and healthy, will mate, a bit less willing if the male is too eager, but any time of day or season.
If you artificially incubate the eggs, you could hatch pretty much year round.
Hens often lay less frequently in the shorter days of winter. Their eggs are not less fertile, but there are simply fewer eggs to gather, which reduces the amount of chicks you can hatch simply due to the fewer eggs.
A rooster (including your Red Jungle Fowl) can mate once with a hen, and she will remain fertilized for at least 2 weeks, sometimes longer. She has a pouch inside that hold the sperm. As eggs develop, they are fertilized by the sperm in the hens little pouch.
If you have an especially fluffy type hen or rooster, then the feather fluff can reduce fertility simply because the vents of the rooster and the hen do not meet fully. (A rooster does not have a male appendage to insert into the hen).
So there is no season to create a more fertile male. Maintaining good overall health in your roosters and hens will make for fertile eggs.
Also, using breeding stock from age of maturity (6 to 8 months) to 2 years of age will ensure better results. Many roosters remain fertile for up to 5 to 7 years of age, though slowing obviously.
Hens are most fertile from point of lay until 2 or 3 years of age. Often you see egg quality issues in the older hen as well as less frequent laying. However, if the older hen is still laying good eggs, likely they are fertile.
You can check fertility of the eggs by cracking one open. You should see a bullseye on the yolk indicating a fertilized blastoderm (the hen's genetic spot on the egg).
Fresh eggs are best to incubate. Fertility in the egg is reduced the older it gets. Set eggs within 7 days of laying for best results.
HTH
LofMc