Game of Thrones Season 8 is not taking the path fans had hoped, which, considering the last seven seasons, should come as a surprise to no one. After so many years of believing Daenerys would come to Westeros and clean up the mess of political nonsense in King's Landing and "break the wheel," she has instead lost two dragons and looks to be on the verge of losing the Iron Throne as well. This has driven fans to extreme theories of a deus ex machina to save the Mother of Dragons. Does Drogon have children? That's the newest magical way out being considered.
It is possible that Drogon could suddenly lay a clutch of eggs. Author George R.R. Martin hasn't been forthcoming on dragon reproduction in A Song of Ice and Fire, but there are hints of asexual reproduction. For instance, dragon mating rituals are never discussed, which, considering the scale of Martin's world building, is a sign they might not exist.
Martin merely says when a dragon lays eggs, it can be a surprise, revealing an assumed male dragon to be female. It never says dragons change their sex, but it unlikely multiple riders grossly misgendered their dragons, unless sequential hermaphroditism was going on.
However, the theory Drogon will suddenly lay eggs is problematic, because of what fans are basing it on.
Game of Thrones Season 8 didn't just increase the budget for battles, costumes, wigs, and filming locations. The production also upgraded the opening titles. The maps have always changed season to season (and sometimes episode to episode) depending on where characters are. But this was a full-on overhaul, showing interior locations for the first time. Also, the show updated the carvings seen in the astrolabes. Instead of scenes from Westerosi history, which occurred before the start of the series, the astrolabes now show more recent historical events.
The original astrolabe carvings showed Aegon's conquest, Robert's Rebellion, and The Baratheon Coronation. These were shown in a hieroglyph-type form, with images of the House Sigils standing in for the men who fought. So Rhaegar vs. Robert at the Trident is a Dragon fighting a Stag, Robert's coronation shows a Lion, a Wolf, a Kraken, and so forth, bowing to the Stag.
The new images follow this design. The Red Wedding, for example, shows a wolf dying at a party with a Lion watching.
And then there's this one:
The image is of a giant dragon and three smaller ones surrounding her. There's a red comet blazing overhead. In desperation to find proof Drogon will lay eggs, fans have latched onto this, insisting it's not a historical moment from the series, but a moment from the future. The giant dragon is Drogon, with three brand new babies.
It's a nice idea, but that's likely not what this is. This is the end of Season 1 when Daenerys births the dragons. The red comet seen in the sky is the one everyone remarks on in the Season 2 premiere. The giant dragon represents Daenerys and House Targaryen. The three smaller ones are Drogon and his late brothers Rhaegal and Viserion.
Drogon may magically lay eggs next week, but this image isn't proof of it. Sorry guys.