Learn Italian: Non abbiamo intenzione di cambiare idea. - We do not intend to change our minds.

Because the person having the intention (we) and the person changing their mind (we) is the exact same. Whenever the subject is the same for both actions, Italian drops the che + subjunctive structure and simply uses di + the infinitive verb (cambiare).

Could I still use che and say Non abbiamo intenzione che cambiamo idea if I wanted to?

No, you should avoid that. In Italian, when the subjects of the two actions match, using di + infinitive is considered mandatory, not just an option. Using che when you don't have to sounds very clunky and unnatural to a native speaker.

Why does the translation say "our minds" when the Italian just says idea?

The English idiom "to change one's mind" translates to a different idiom in Italian: cambiare idea (literally "to change idea"). You don't use the literal word for mind (mente), and you don't need to add a possessive word like "our" (nostra).

Why is there no article before intenzione? Shouldn't it be l'intenzione?

Avere intenzione di (to have the intention of / to intend to) is a fixed expression. Just like with avere fame (to have hunger / be hungry) or avere bisogno di (to have need of / to need), the article is dropped.