This is a regular verb with a spelling change. It follows the conjugation pattern of the verb vencer.
PresentPresente
Sus argumentos siempre me convencen.
His arguments always convince me.
PreteritePretérito indefinido
Al final convencí a mi padre para viajar.
In the end I convinced my father to travel.
FutureFuturo
Sara convencerá al jefe con su plan.
Sara will convince the boss with her plan.