Neither convenzo nor convezco exists in standard Italian.
"No one leaves this room until I get the truth. I have done this a hundred times. I even the most innocent-looking people to admit their guilt."
So, why the confusion? The reason lies in the similarity in pronunciation and the fact that both verbs are conjugated in a similar way. Additionally, in some Latin American countries, particularly in Central America and the Caribbean, "convezco" is sometimes used as a colloquialism or a regionalism. However, it is essential to note that this usage is not universally accepted and may not be considered standard Spanish.
Here is the explanation and a complete story to help you remember why.
To remember the spelling, just think of the similar verb (to conquer/defeat).