Mercutio certainly does show impulsive behavior. For the most part, we see him act impulsively the day Tybalt kills him on the street.
Mercutio had absolutely no reason to be out on the street that day. In fact, Benvolio begs him to come inside with him, warning
The day is hot, the Capulets abroad
And if we meet, we shall not scape a brawl,
For now, these hot days, is the mad blood stirring. (III.i.2-4)
The only reason we have from which to speculate why Mercutio is out on the street is that he is angry because he just learned that Tybalt has sent a letter to Romeo's father and believes that Tybalt has challenged Romeo to a duel (II.iv.6-8). Hence, we can speculate that Mercutio is out on the streets because he truly is hoping to pick a fight with Tybalt or other Capulets. Since Mercutio is acting purely from emotional response and without good sense or judgement, we can say that he is acting impulsively in staying out on the street.
Meructio further acts impulsively when he treats Tybalt insultingly, even though Tybalt is actually very civil. Tybalt civilly greats them as gentlemen and wishes them a good evening in the line, "Gentlemen, good den," and even asks permission to speak with them (III.i.38). Mercutio, responding out of pure emotion, returns Tybalt's civility with a threat, saying, "make it a word and a blow," meaning a blow from their swords (41).
Finally, when Romeo refuses to fight Tybalt, Mercutio calls Romeo's refusal "calm, dishonourable, vile submission!" and challenges Tybalt himself, leading to his own death, Tybalt's death, and indirectly, Romeo's own death (71-80).
Hence, we clearly see from this scene that Mercutio responds purely from emotion rather than from rational thought, thereby acting impulsively.
Yes, Mercutio has impulsive behaviors, such as when he instigated a fight with Tybalt, eventually leading to both their deaths.