A billion year old technologically advanced civilization in our own galaxy would probably be fairly easy to find as current models estimate that it'd only take a couple million years to colonize the galaxy even without FTL travel. Thus we'd expect to see various technosignatures scattered everywhere (including within our own solar system).
They'd have to either have advanced extremely slowly compared to us, such that a thousand years of our development are many millions of years of development to them or had some sort of extinction event that was so thorough as to render all of their stuff around other stars inactive (I guess maybe they might consider it extremely important to hide from developing civilizations though, in which case we probably can't even fathom what sort of hiding mechanisms they might have).
It's worth considering that with another century or two of development, assuming we don't face some sort of societal collapse, we'll definitely be capable of sending small probes to our nearby stars at large fractions of light speed. Given a couple thousand years without a collapse, we most probably can send self-replicating machines to colonize the galaxy like that. Even supposing that it takes us 100,000 years to get to that point, that's just 0.01% of a billion year old civilization's time.