Is there a phrase to say that someone's hidden intentions are revealed in his/her talk or movements?

I would say:

he's showing his true intentions

or use the following idioms:

show somebody's true colors
reveal somebody's true colors

In gambling circles, this is referred to as a "Tell". For instance, if one poker player notices that another player scratches his face when he is nervous or bluffing, the first player has identified the second players "Tell". This is a slang term that refers to something the second player is doing that "tells" a different story than the one he wants you to believe. A "Tell" can be a nuance in body language, a gesture, a tone of voice, or any other subtle indicator one person has identified in another. It's a signature action that is "telling" you something contradictory to what the person is actually saying.

The closest English phrase in form is

His mask is slipping.

Je ne connaissais pas cette locution, mais en anglais c'est peut-être Truth will out.