yes
Tybalt was asking for a fight with Romeo. Since he was married to Juliet and Tybalt was now his cousin, he refused to fight him. Then Mercutio asked to fight Tybalt and in a way kind of tempted him to fighting. Tybalt ended up killing Mercutio because Romeo got in the way when trying to stop the fight between the two. So now Romeo is angry that his friend Mercutio is dead and Tybalt doesn't have a scratch. So, Tybalt and Romeo fight and Romeo kills Tybalt.
The immediate outcome of the fight between Romeo and Tybalt was that Romeo won and Tybalt died. This had the effect of getting Romeo banished from Verona.
earlier in the play, romeo crashed a Capulet party. angry, Tybalt swore revenge with a sword fight. Romeo's friend mercutio ends up fighting Tybalt, and romeo attempts to intervene by stepping between them. Tybalt stabs mercutio, despite Romeos attempts to end the fight, and mercutio dies. Romeo and Tybalt then fight, and romeo kills Tybalt, and romeo is banished from Vienna. Merutio, romeo and Tybalt sword fight.
The devil is to blame. Really.
Mercutio is the one who decides to fight Tybalt.
The fight between Tybalt and Romeo takes place in Act 3 scene 1 lines 113-121 (Cambridge Schools Shakespeare)
Tybalt wants to fight him. Tybalt always wants to fight someone, and he has fastened on Romeo after Tybalt spotted him at the party in Act I Scene 5.
He calls him a villain. A pretty lame insult, and not enough to get Romeo to fight. It is enough to get Mercutio to fight, however. Killing Mercutio does for Tybalt what no insult could do--it gets Romeo to fight him. Unfortunately it does for Tybalt in another sense altogether.
Romeo refuses to fight Tybalt. Mercutio steps in for Romeo and fights Tybalt. Romeo steps between them, trying to stop the fight. Tybalt stabs Mercutio. Mercutio curses both the Capulets and Montagues. Mercutio dies. Romeo is enraged and decides to fight Tybalt. Romeo kills Tybalt, then he runs. Prince Escalus comes to the scene. He announces Romeo's banishment.
At first, Romeo does not want to fight with Tybalt because he is now married to his kinswoman, Juliet. This makes Romeo and Tybalt of the same family, and Romeo does not want to fight his own.
Romeo kills Tybalt.
Basically they were spoiling for a fight. Tybalt was looking for Romeo in order to fight him, but Romeo didn't want to fight Tybalt. Mercutio didn't like Tybalt at all and was looking for a chance to fight him. He used the situation as a pretext to challenge Tybalt, who was not a man to turn down anyone's challenge.