Why did axel not want to go on the trip?

Journey to the Center of the Earth is a novel by Jules Verne, originally published in French in the year 1864. The story is narrated by Axel Lidenbrock, a young man who lives in Hamburg, Germany with his uncle Otto Lidenbrock, a professor of mineralogy.
Axel assists Otto with his experiments, and definitely takes after his uncle in terms of having a scientific mind. As he says himself:

I will admit that I devoured geological science with great relish; I had mineralogist’s blood in my veins, and never felt bored in the company of my precious pebbles.

When Otto finds a manuscript written in Icelandic runes in a shop, he takes it home, and a separate piece of parchment with a message jumbled in a runic code falls out. Axel, almost by accident, figures out how to decode its secret message before Otto does. Axel reads it and finds that the message describes a way to travel in order to reach the center of the earth. Knowing immediately that his Uncle will want to take this journey, he says the following:

‘No! I’m not going to tell my uncle. It would be terrible if he got to know about such a journey. He’d just want to have a go himself. Nothing would stop a geologist of such determination. He would leave anyway, against all obstacles, whatever the cost. And he’d take me with him, and we wouldn’t come back. Never. Not nohow!

Later on, after Axel eventually gives in and tells his uncle how to read the code, and Otto declares that they're going to go on this journey, Axel tries to reason with him, saying:

All right, I accept that this Saknussemm wrote the message, but does it necessarily follow that he actually carried out the journey? Couldn’t the old parchment just be a practical joke?’

And so, we see that Axel does not want to try to go to the center of the earth because he doesn't fully believe that the feat is possible, and thinks that they would die trying.
(Note that any quotes I used in this answer may be slightly different from the ones in your own copy of the book, as there are multiple different English translations of Journey to the Center of the Earth.)