Axel does not want to go on the trip because he is, in a word, frightened. The journey seems to him to be fraught with danger, what with having to go inside a volcano (albeit an extinct one) and deep underground.
Axel conveys his emotions vividly as he ineffectually attempts to persuade his uncle Lidenbrock to turn back. However, Lidenbrock is inexorable. He is eager and willing to take any risk in his pursuit of scientific exploration. Indeed, he behaves as though there were no risks at all, for example when he refuses to take along enough water for the journey, counting on underground springs to see them through. Axel is dismayed at this:
The fears I had expressed as to the quality and temperature of these springs, and indeed as to their existence, had been totally disregarded (chapter 11)
Lidenbrock, then, refuses to listen to Axel, the voice of reason. Axel is taken aback at such a dismissive attitude, and his fears are borne out in this case when they nearly die of thirst at one point during the journey.
His uncle’s irresponsible attitude increases Axel’s qualms about the whole expedition. However, Axel learns how to face up to his fears and sees the journey through to its unexpectedly successful conclusion. Indeed, he, his uncle and their accompanying guide Hans all return as heroes.