Learn Italian: Ribadisci che non vuoi pagare la tua quota? - Do you reiterate that you do not want to pay your fee?

If the person were speaking directly, they would say, "I don't want to pay my fee" (la mia quota). Because we are repeating their statement back to them as a question ("Do you reiterate that you..."), the possessive shifts from la mia to la tua to match the new subject.

Why is the verb vuoi used instead of voglio if the person was originally talking about their own fee?

This is another necessary shift for indirect speech. The original direct quote was "I don't want..." (non voglio). Since we are now addressing the person directly about what they said, the verb shifts to the "you" form (non vuoi).

Is ribadisci an irregular verb?

No, it is a regular -ire verb, but it belongs to the group that adds -isc- to the root in the present tense, just like capire (to understand) or finire (to finish). The infinitive is ribadire, so the "you" form becomes ribad-isc-i.

Does quota only mean "fee"?

It generally translates to "fee," "share," "quota," or "portion." In this context, it refers to a person's expected financial contribution, like a club membership fee or their specific share of a group bill.