There is no way that I would know the answer to that question.

[GoldfishLord]

(I) wouldn't know.​

There is no way that I would know the answer to that question. John: When will the flight from Miami get in? Jane: Sorry, I wouldn't know. Bob: Are there many fish in the Amazon River? Mary: Gee, I wouldn't know.

Source: https://idioms.thefreedictionary.com/I+wouldn't+know

Does "would" mean something to similar to the "would" in "if some condition were met, I would have the information"?
What does 'would" mean?

 

[Tarheel]

Jane doesn't know when the flight from Miami will arrive. There is no reason she would know. She doesn't have that information.

Mary doesn't know if there are a lot of fish in the Amazon. (There probably are.) She doesn't feel qualified to answer that question.

Questions about "would" are seemingly endless.

 

[jutfrank]

I think the best way of explaining this is that I wouldn't know means 'I don't know, and I'm not a good person to ask anyway'.