House of the Dragon has fans flying seahorse banners and cheering for House Velaryon, though they were never shown in Game of Thrones. Some viewers have even observed the dark implications of Westeros' only noble house populated with Black people being absent from the main series. However, George R.R. Martin's books offer some explanation.
Fair warning: there are spoilers ahead for A Song of Ice and Fire! House Velaryon rules the island of Driftmark from their castle, High Tide. Most fans are familiar with the island of Dragonstone in Blackwater Bay, but without reading the books you might not realize that there are other islands in the bay that are vassals to Dragonstone. According to the books, the Velaryons have a long lineage going back to Old Valyria, but they settled Driftmark generations ago – long before the Targaryens came to Westeros. Before allying with House Targaryen, the Velaryons were not dragon-riders, but through marriage, some of them became dragon-riders in Westeros.
In House of the Dragon, House Velaryon is wealthy and powerful, but their fortunes rise and fall like everyone else. According to Martin's "imaginary history book" Fire & Blood, Driftmark amassed immense wealth thanks to Lord Corlys Velaryon, who made nine legendary trading voyages in his youth and brought wonders from around the world back to Westeros. Another spinoff series about these voyages is in development at HBO.
Obviously, Corlys and his family remain active during the civil war and the time period in which House of the Dragon is set. Without spoiling the ending of that show, suffice it to say that House Velaryon endured as an institution and retained its influential status. There are then about 170 more years before the beginning of Game of Thrones, some of which is described in Martin's history. Alliances are made and broken, but things don't really get relevant to the main story until the war known as Robert's Rebellion, where Robert Baratheon cast down "The Mad King" Aerys II Targaryen.
House Velaryon remained loyal to the Mad King during this time, with its then-lord Lucerys Velaryon serving as the Master of Ships for him. After House Baratheon won the war, the new King Robert made his younger brother Stannis the Lord of Dragonstone, and tasked him with managing vassal houses like the Velaryons. They remain loyal to Stannis durin the main story, providing ships and warriors for his side in the War of the Five Kings.
HBO's Game of Thrones did not portray any Velaryons on-screen – or many of Stannis' other vassal lords, for that matter. However, in the books we learn that the hawkish Lord Monford Velaryon dies in the Battle of the Blackwater, burning alive when Tyrion Lannister ignites wildfire on all the ships. He is succeeded by his 6-year-old son Monterys Velaryon, who continues his house's support for Stannis.
In the books, Stannis is still alive and fighting in the north, and as far as readers know the Velaryons are still among them. Jon Snow observed their banner among Stannis' camps in book three, A Storm of Swords. Meanwhile, in book four, A Feast For Crows, a bastard of House Velaryon named Aurane Waters joins Cersei and becomes her Master of Ships. However, he betrays her and runs off with several ships, establishing himself as a pirate on the Stepstones islands.
As you can tell, House Velaryon is present in A Song of Ice and Fire and, presumably, in Game of Thrones – they're simply not as powerful or influential as they were during the time period of House of the Dragon. Hopefully, HBO will greenlight its prequel The Sea Snake so that fans can learn more about House Velaryon in the years to come. In the meantime, Game of Thrones and House of the Dragon are both streaming on HBO Max, while Martin's books are all available in print and digital formats.