"We found a dozen faked invoices hidden in Jim's desk; these invoices put Jim's integrity in doubt". That's how I would usually use this; when some finding seems to show evidence that Jim might not be an honest person, that puts Jim's honesty or integrity in doubt.
"I talked to Jack who made some interesting remarks about Jim, and now I doubt Jim's integrity". I doubt his integrity means I personally believe that Jim is not honest.
"I think I caught him lying, so I question Jim's integrity". This means it's more than just a personal belief, I am making a suggestion for everyone to hear that Jim might be a dishonest person.
The question "Do you really put my integrity in doubt?" would therefore be a bit strange. "Does X put someone's integrity in doubt?" means "Is X really evidence about someone's lack of integrity?" Much more usual would be "Do you really doubt my integrity?" meaning "Do you really think I am a dishonest person?" or "Do you really question my integrity?" meaning "Are you really suggesting to other people that I might be a dishonest person?"