If I doubt THAT you told me the truth, I am reasonably certain that you told me a lie.
If I doubt WHETHER you told me the truth (or not), I am more uncertain: you are probably lying, but there is still a real possibility that you're telling the truth.
Since you're also learning French, note that "se douter" would translate as "suspect," NOT doubt.
Ils se doutent que je leur cache quelque chose.= They suspect that I'm hiding something from them.