There are 4 ways:
1) Have ambition to get the egg. You need to have at least 800 gold for event to fire, but MTTH is about 7000 years IIRC, with some traits to reduce it. This is unlikely to happen, though, because of such long MTTH.
2) Get it from Smoking Sea. You have 1% chance and for event to fire, you can´t have Valyrian steel sword or more than 500 gold. Good luck with that. There is also 1% chance for sword and 3% chance for 500 gold...and 95% chance of starting to worship Cthulhu (which kind of makes it win-win situation, honestly). This can also fire only once per every character, so chances are also extremely small.
3) Ruins of North Valyria. Howling Bogs (Telyria), Draconys, Anogaria and maybe Bhorash (or how is that ruin on eastern part called) can give you egg. Colonize them and then pray to gods of RNG, because MTTH is 120 years. And they have to be colonies or advanced colonies to give you an egg, so you can´t just hold those provinces until you get the egg. And you can get only one for each province.
4) Summerhall after year 8259. This is your best bet. MTTH is only 60 years and you can get that event several times (although only once per character).
You could always use ruler designer, though, or open console and type: add_trait dragon_egg, and it will give it to your current character.