"The cow's calf is there in the garden, next to the big cow."
Translation:A tehén borja ott van a kertben, a nagy tehén mellett.
No, you don't want the possessive suffix -e on the cow. Your sentence would be "His cow's calf is there in the garden, etc."
Your options for the cow are either the dative -nek ending, or no ending at all. A tehén borja ott van... would also be good.