Book about party quest involving a dragon egg

I believe the book in question is probably older because I think I found it in my grandparents stacks when I was younger. I’ve scoured them several times since and couldn’t find it but they have a lot of turn over. It definitely seemed older due to the condition of the book and the paper if I’m remembering correctly. At the very least I don’t think it was a book specifically written with children in mind as the target audience.

Anyways, I think the book centered around a young man or older boy who I think found a dragon egg. This was the last egg or there hadn’t been dragons seen in a long time so a party was gathered to take this egg to the hatching grounds or something like that. The only specific I can remember for sure is from one scene where the party is climbing a mountain. There was specific imagery of the party members being tied together with the large knight type guy at the bottom and the spindly main character leading the way.

I remember something about concern that if the knight was to slip the rest of the party might not be able to support his weight and if the young man at the top slipped and knocked the other central party members loose the knight guy might not be able to catch them all.