In the context of the Trojan War, the Trojans boast that "Ilium will stand forever" because they have strong faith in the invincibility of their city, Troy (also known as Ilium). The Trojan people believe that their city is protected by the gods and that its walls are impenetrable. They have withstood many attacks from the Greeks during the war and have successfully defended their city.
The Trojans' confidence and pride stem from their belief in the divine protection of their city, especially with the presence of their patron god, Apollo. They view the Greeks as mere mortals who cannot conquer their great city. The long siege and the failures of the Greek army to breach Troy's defenses reinforce the Trojans' belief that their city is indestructible.
Additionally, the Trojans have a sense of pride in their lineage and heritage. Troy is considered the home of the noble Trojans, descendants of the gods, and they take great pride in their ancestry and the rich history of their city. This pride fuels their conviction that Troy will endure and remain unconquerable.
However, this boast of Troy's eternal standing ultimately proves false, as the Greeks manage to infiltrate the city through the famous Trojan Horse, leading to its eventual downfall.