So if we went at the max warp (warp 9.975 or 3,053 times the speed of light), we could be there in 20 million /3053 = 6551 years.
Can we use replicator machines to build new star ships to house the surplus population growth while we travel?
Assume we start with a crew of 1000 people (Enterprise was 1014 give or take a red shirt or two).
And assume we grow the population at a 1% annual growth rate (Historical human growth rate peaked at 2%). They have advanced health care, won’t run into many hostile aliens in intergalactic space, and presumably can avoid civil war.
We would arrive in 6500 years with a population after 6500 years of compound growth at 1% of ≈1.2×10^31.
And we would have built 1.2x10^28 enterprise class star ships to hold that population.
How big of an invasion force do we need for a whole galaxy?