Can I season firewood over the woodstove?
Proper seasoning is going to take longer than that. Most experts recommend you buy green firewood in the spring, to let it season in the rack over the summer. A minimum of 6 months' drying time, leaving it exposed to the sun and protected from moisture, is needed to "cure' firewood.

DO NOT burn your green firewood. It will smoke, smoulder and not heat the chimney properly for good draft, so you'll get backdrafts into your home. I had this problem even with seasoned but dampened logs from my rack outside; the wood wouldn't burn properly even with a nice hot bed of coals under it, and we got a lot of smoke coming out the front of the fireplace (and up into the upper story of our house).