BELIEVE IN the notion or BELIEVE the notion?

When I don't trust what someone says, then I need to say,

I don't believe you.

without 'in'.

But for the notion, it's a concept, so I think I need to say,

I don't believe in the notion.

but on Google, I could find a lot of results saying "believe the notion".

Is it possible to use both of them?

They mean different things. To believe something is generally to accept it as truth. To believe in is generally either to accept that something exists (believe in Santa Claus) or to align yourself with some ideology (believe in capitalism). People sometimes use the "ideology" sense with real people, however, leading to the nominally nonsensical I don't believe in Donald Trump, which seems to question his existence but actually just disavows his policies.

So, you probably believe the notion that there are cars that use petrol (gasoline) and diesel in their internal combustion engines, but you may nonetheless believe in the notion that they should be replaced with vehicles using alternative fuels.

The Merriam-Webster dictionary has a clear definition. It says:

Definition of believe in 1: to have faith or confidence in the existence of (something) Do you believe in ghosts? 2: to have trust in the goodness or value of (something) She believes in (the value of) regular exercise. I believe in working hard to achieve success. She doesn't believe in using pesticides. 3: to have trust in the goodness or ability of (someone) Despite his problems, his parents still believe in him.

That is clear, I think. Belief in is a kind faith or trust

If you look at definitions of believe (without the preposition), they all involve thinking that some statement or collection of statements is true, or that the person making the statement is telling the truth (ie that a statement that person is making is true.

There is one important exception to this. If you say "I believe in Marxism" or "I believe in Buddhism". Sometimes it means that the speaker believes that something exists: God, black holes, etc. Sometimes it means you trust or agree with something: democracy, President Trump or what or whoever.