Converting to mongol culture wouldn't start the mongol invasion. You can sort of become the mongol invasion by creating the mongol empire (which is a decision requiring you to hold a bunch of land as a mongol nomad ruler) but this disables the massive event troop invasion. You have to conquer it all with your basic horde troops, which fortunately once you're that size is not particularly hard.
The thing is, creating the Mongol empire requires you to be a nomad. Feudal or tribal mongol characters can't do it. Nomads also can't change culture (I experimented with this quite a lot when horse lords came out, it is possible to work around and get an heir of a different culture, but the game simply disinherits them meaning your game will end unless you have appropriate culture children). This means the only way to become the Mongol empire is to start as a Mongol culture nomadic character (like the Bayad clan in the Charlemagne start).
As a feudal character, you can certainly convert to Mongol culture (or Khitan culture, which is effectively treated the same by the game) and it's really good (mandatory agnatic succession means very stable realms, you have free raiding and, if you're pagan, you can launch invasions) but it won't have any impact on the mongol invasion itself, except that when they do show up they will obviously like you a bit more for being the same culture.
