Japanese will not distinguish nouns like English does. `A` and `The` are both English particles that _do not exist in Japanese_. If you are looking for a linear translation, you won't find it.
In both cases, I would translate it as:
>
If I were to talk about a specific fish, then I would add more information to let the listener know what fish I ate. For example:
>
> I ate the fish that was in the fridge.
>
>
> I ate that fish.
As you study more Japanese, you will realize that particles in English and Japanese rarely line up, as is the case with the two you are asking about in this question.