"Wir glaubten seiner Aussage nicht" is this correct?. The resource uses here the dativ, while normally it shoukd be "etwas (akk)glauben".
The dictionary says that glauben can come with etwas in dative case. That is correct. And seiner Aussage is in dative case. So, the dictionary gives a correct example for a correct rule here.
The dative option seems to be surprising to you. I think you can understand the dative by conceptualizing the thing that is believed in (etwas in dative case) as something like a person. The non-personal Aussage ("statement") is being personalized. As in persons, you can trust in it or not. It is as if the thing that is believed in or not is thought of as a thing that is speaking on its own, as a person.
So, it is possible to use the accusative (Wir glaubten seine Aussage nicht.) or the dative (Wir glaubten seiner Aussage nicht.).
The difference in meaning between using dative compared to accusative case is very subtle, if there is any. Although the sentence in question (with dative case) could be translated as
We did not believe his statement.
it would more appropriately translate as
We had no confidence in his statement.
as the latter translation would transport the personalization of statement a little better.
glauben kann simply take two optional objects:
Here's a sentence that has both:
Wir glaubten ihm seine Lügen nicht mehr.
The dative object denotes the "owner" of whatever is to be believed - He is the one making the claim, typically a person or personalised object.
The accusative object is the thing that is to be believed in, the "claim" - here, it is "his lies". Not a person, typically.
Wir glaubten ihm nicht mehr
This only has the dative object, the one making the claim. What it actually is, is not said.
Now it is getting a bit tricky. In our specific example, the lies can be either dative or accusative
Wir glaubten seinen Lügen nicht mehr (1)
Wir glaubten seine Lügen nicht mehr (2)
Even if the basic meaning of both sentences is practically identical, there are (very) subtile differences. (1) Could be constant (new) claims being made, while (2) could be a fixed number of claims that we simply no longer believe.
jemandem (Dat) etwas (Akk) glauben.
The Dativ object refers to the person or thing you trust in or believe in.
The Akkusativ object refers to what you trust to be true, happen, etc).
The sentence "Sie glaubte seiner Aussage nicht" is the construction (5) in your reference which is Dativ, the object which describes whom or what is trusted in.
Ich glaube Dir Deine Worte
I trust you, and that your words are true.
We can expand your original sentence with an additional Akkusative object:
Sie glaubte seiner Aussage (nur) den Teil, der sich auf die Goldfische bezog.