Answering from an Indian perspective, so might or might not be relevant/suitable to professional environments elsewhere:
If I go and tell him that I don't believe him, what are the possible
situations I may face ?
You can get fired. If not fired, you'd just demote yourself on his list of favourites.
In India, the hierarchy is very rigid, and as you go higher up the tree, you get tougher managers with higher ego.
So, confronting him should be the last thing you would want to do.
Break it to him as softly as you can. You can send him a polite meeting invite (like sort of a 1-on-1) and inform him:
I am really enjoying work under your mentorship. However, there is a proposal which I want to make so that both of us can keep track of my goals and targets properly.
It would help me track my progress and thus would help me improve my career, and the quality of my work at the company. It would also help you gauge the work of our team and help us document the progress and achievements, which would help us in meetings and also project demonstrations.
And then, propose the solution, be it a e-mail documentation, or a software like jira, or keeping a hard copy of the projects(<-- not recommended!)
Make the proposal look as if it would benefit him more than anyone else.
So, if you can't confront him, convince him instead.