So before we start we should clear something up, since it's pretty important to how we go about answering the question. There is no First Triumvirate. The term is a convenient fiction that classicists use to describe the extremely complex relationship between Caesar and Pompey around 60, B.C., to which Crassus was in some way privy but not really a direct beneficiary. Really there was only one Triumvirate, the one where Mark Antony, Lepidus, and Octavian all received a triple extraordinary magistracy legally by the senate. This is reflected in Latin literature, which only refers to a single Triumvirate and only three triumvirs. The identification of the "First Triumvirate" is something much later, mainly as a result of the Victorian scholars' willingness to buy into Augustan propaganda, which tried to legitimize the Triumvirate by setting up a precedent. Classicists still call it the First Triumvirate because it's convenient, but not really accurate.

In reality the political ties and tensions between Caesar and Pompey were much more complex. They never really formed an alliance, unofficial or otherwise, which is part of the reason why it's not really correct to refer to a First Triumvirate. The so-called First Triumvirate is usually identified as a single event, the agreement between Pompey and Caesar that Caesar would not oppose Pompey's Eastern Settlement in exchange for a better province for his proconsulship. Crassus was there, too, but he didn't really benefit that much, only agreeing to pay off Caesar's debts in exchange for Caesar's support in politics, which was meaningless (Caesar had already usurped control of Crassus' dying Marian party) and which in any case he never really got (until his death in Parthia Crassus does absolutely nothing of any importance). The reason that the First Triumvirate is dated to this point, despite the fact that Caesar and Pompey had been maneuvering around each other and often working together (after all, Caesar had started his career when he returned from his self-exile in Asia by supporting Pompey when no one else would) because it's the first and only time when the three men are known ever to have met each other at the same time. Ever. Now the exact nature of their political ties is a subject of heated debate, since it depends very heavily on the nature of Roman political and social trends between individuals, which is hotly debated and which we don't yet fully understand. But what's very clear is that this is very much a one-time deal. As soon as Caesar left for Gaul Pompey was already openly undermining Caesar's position in Rome. For example, Clodius Pulcher's (an asset of Caesar's who was difficult to control and whom Caesar tried several times to distance himself from) assassination in 53 occurred while Caesar was away in Gaul and unable to defend Clodius in court, leading to Milo's light punishment which infuriated the Caesarians. Pompey's troops and support for Cicero was instrumental in the affair.

So what role did Julia play in the affair? Julia's marriage to Pompey should be seen rather like sealing a contract. What's very clear is that the deal was sealed between the two of them. What's not so clear is what their relationship was supposed to be like after Pompey's Eastern Settlement went through and Caesar got his proconsulship in Gaul. Generally it's assumed that Julia's marriage was supposed to be a way of some sort to prevent direct confrontation between the two of them, although to what degree is pretty unclear. Certainly after Julia's death Pompey, who had been slumbering, possibly biding his time, while Caesar was away, suddenly lashed forth in a whirlwind of political activity, backing Cicero and the Milonians, and throwing his weight around much more than he had in the past few years. Why this is is also unclear. The textual material suggests that Pompey was quite devoted to Julia. For example, although made governor of Hispania Ulterior he refused to go, governing through his subordinates, supposedly because he didn't want Julia to have to leave the city. Plutarch claims that Julia was devoted to him and vice versa, and that it was even rumored during the period from about 60 to her death that Pompey had given up on politics. It's also possible that Julia's death may have absolved Pompey from any duty, real or imagined, to support or at least not openly come into conflict with her father.

Julia's death is probably more important as a marker than as any sort of cause. Undoubtedly it was the cause of something (Pompey was, after all, human, and Caesar supposedly never got over Julia's death), but what that was is difficult to pin down. What's very clear is that it marks the point at which Pompey woke up from his five-year slumber. Pompey prior to 60 had gotten everything he wanted except the Eastern Settlement, which Caesar secured for him. His activities up to 54 are slow and uninspired. He seems to have been attempting to consolidate his position, but didn't really do much in those years. All of a sudden he wakes up after Julia's death, siding with Cicero and in 53 being the instrumental force in making sure Cicero gets Milo off clean after his murder of Clodius. It's a very sharp change, to suddenly stab Caesar in the back after five years of inactivity, and many ancient observers theorized that Julia was at least part of the reason for it

So the short answer? Eh, it's difficult to say. She certainly couldn't have torn down the First Triumvirate, since there was no such thing, but certainly her death soured the already uneasy "understanding" that Caesar and Pompey had. To what degree can certainly be debated.