In Act 3, scene one, Tybalt's argument is with Romeo. He is not interested in fighting Mercutio. Tybalt wants to fight Romeo for his embarrassment of Romeo's appearance at the Capulet's masquerade party.
Tybalt wants revenge because Romeo crashed the party. Tybalt has no idea about Romeo's marriage to Juliet at this point.
Romeo does not want to fight Tybalt since he is now his relative. Mercutio begins to fight Tybalt, embarrassed by Romeo's submissive nature. Romeo gets in between them and Tybalt kills Mercutio, stabbing underneath Romeo's arm.
Romeo must kill Tybalt to avenge Mercutio's death. Now Romeo must flee the scene to keep from being sentenced to death by the Prince.
Tybalt did not want to fight Mercutio because he was only seeking revenge on Romeo for coming to Capulet's party in act 1. Tybalt only wanted to duel Romeo and didn't want anything of Mercutio.