I Henry IV is peppered with broils. Most of them take place off-stage. Act I Scene I opens with the King talking about the recent end to the civil wars and his plans to immediately arrange for a crusade to the Holy Land. Then Westmoreland comes in and tells him about a battle that had been fought in Wales the night before between the "irregular and wild Glendower" (a Welshman) and the Percies (Hotspur, his father, and Archibald), wherein Hotspur took several valuable noble prisoners. If by "the first fight," you refer to actual battle, this is the one.
The term is ambiguous, however, because there are fights between the main characters. First, the King demands that Hotspur turn over his spoils of war (his noble prisoners, so the king may ransom them) and Hotspur refuses, thus precipitating the Battle of Shrewsbury, the only battle that takes place on stage in this play. Also, because Hotspur loves to fight and apparently doesn't really care who he's alienating, there's a heated argument between him and Glendower on the eve of the battle (Act III Scene 3).