you need to find the carrot seeds in the black forest. The carrot plant has white flowers. So when you equip the cultivator with the seeds in the inventory the option to plant carrots will appear.
With the cultivator equipped you need to plow the land first and then click on the carrot option and click on the soil to plant.

Plant separately otherwise your carrot dies.
Also, it cannot be planted in a place covered by a roof or under trees.
You must also protect your crops from the mobs they kill your plants.
After your carrots grow you can plant them again to multiply the seeds.