"don’t you believe it!" (informal) used to tell somebody that something is definitely not true. ‘She wouldn’t do a thing like that.’ ‘Don’t you believe it!’
would you believe (it)? (informal) used to show that you are surprised and annoyed about something. And, would you believe, he didn't even apologize!
I don’t believe it! (informal) used to say that you are surprised or annoyed about something. I don't believe it! What are you doing here?
So, it seems that "would you believe (it)?"="I don’t believe it!"
But I am not sure what "would" in "would you believe (it)?" means? or it is just a way that native speakers say without any particular meaning?