Actually, there is no specific season for spearfishing. However, spring and fall seems to be the best time for spearfishing. For instance, water is less tense during spring which makes it excellent for spearfishing.
There is a wide variety of fish around the river to catch at the season of fall which makes spearfishing easier. Also the temperature falls during this period and the night becomes longer.
Spearfishing is excellent in the afternoon to early evening, but, poor in the late morning to early afternoon, and the winter is not your best spearfishing season, unless, it is ice fishing and it is quite different from other seasons for it demands different techniques and spearfishing gears.