Wikipedia says it is proven to be impossible. I'm sorry, I'm too lazy to check the proof.

Mathematics is one area, where you don't have to trust the experts, but I would probably have to devote a large amount of time to understand the proof. Mathematics is not it's own field of study with professionals that do nothing else without reason. There is always a level of math that you can only understand by dedicating more time to it.

Maybe I could suggest examining where your intuition comes from, that Hilbert's tenth problem is possible to solve (i.e. there is a certain method to find solutions). Would it have weird implications if it couldn't be solved?

The naive algorithm would be to try different numbers for x, y and z until you find a solution. But you'll never know whether you will eventually find a solution or if you'll keep searching for infinity.

0² + 1 ≠ 0, (-1)² + 1 ≠ 0, 1² + 1 ≠ 0, (-2)² + 1 ≠ 0, 2² + 1 ≠ 0, ...