The thing is, out of a population of trillions (or even quadrillions as you say), you only need a few thousand to travel to the stars to colonize another planet. With how large the population is, that is bound to happen. Just like there were bound to be pioneers travelling to the new world to settle it, despite how dangerous the journey was. And how there will be pioneers to settle the moon or mars or further out.

And a civilization like that would absolutely send stuff to other star systems, if only for science, so most of the research for the journey would already be done. And this is assuming that a civilization wouldn't want ever greater quantities of resources for ever greater projects, or access to other star systems for reasons we cannot fathom today (maybe neutron stars or black holes have some incredibly tempting uses? Or maybe there's some useful resources out in the galaxy that we have yet to discover?)

Basically, a successful civilization like that is bound to spread out, it's difficult to see scenarios where a successful civilization would be so homogenous in thought as for that to not happen. Amd then it's before we even get to sending AI probes to "colonize" space and gather data.