there are a few ways to go about this, famously there was “Generations Quake 3” if I remember the name correctly. That was a mod for Quake 3 multiplayer that let players choose a character from previous id games and then you would get those games’ gameplay in multiplayer.
I believe id or activision or somebody got legally upset about this and this effort was shut down, this was back way before the quake 3 source code was available and so it was a mod.
For the other way, you’d probably want to check out the RQ3/Reaction Quake 3 work. That is the Action Quake 2 game ported to Quake 3 including various levels made for the quake 2 engine. I don’t know if they used a converter for the levels, but I believe there is software out there that will just convert a Quake 2 level to a Quake 3 compatible format.