You can change Korean Won here in Taiwan. Mega Bank certainly does. And you'll surely get a better rate in Taiwan.
Almost any time NT dollars are involved, it's better to do the exchange in Taiwan. Overseas banks in most places like to tack huge fees onto NT dollars.
As for the banks in the airport, I'd imagine they do exchange Won, but I can't say for certain. You may want to convert enough Won to NT$ in Korea to tide you over for a few days, or if the exchange rate is abysmal (like a 30% loss, which I've sometimes seen with overseas NT$ exchanges), just convert them to US$ and then change the US$ to NT$ at the airport bank in Taiwan.