To begin this quest, talk to the farmer west of Ariela's Tavern. He will take you to the outskirts of the city, through Lazulis Road, and give you your own pumpkin patch to grow pumpkins. He will give you a seedling and fertilizer so you can begin farming.
To grow your own pumpkins, plant the seedling and use the fertilizers.
You can buy seedlings from merchants, so look out for their annoucements. Better quality fertilizer will grow larger pumpkins and garner you a better profit.