"We do not have grapes."
Translation:У нас немає винограду.
No, they are not interhcangeable. The reason why персик and виноград have different case endings is that виноград is uncountable. Виноград is what we call a plant or a bunch of grapes. For the one berry specifically you can use виноградина (plural виноградини)
Also take a look at this post: https://www.duolingo.com/comment/9373116