Yeah, big doesn't even begin to cover it. Some years back I did some quick math and determined that if you could make enough starships so that one arrives at one new star every 60 seconds that would mean 525,960 stars could be visited per year. Wow!
But even at that incredible rate to visit all ~400 billion stars in the Milky Way would take three quarters of a million years (760,514) to accomplish. Even assuming the Milky way is low-end 100 billion stars, and the Alpha Quadrant is ¼ that, it's still gonna take you 47,532 years to drop in on all those stars.
Even if there are civilizations all over, and many of them are gonna contact you, it's gonna take forever to make contact with them all, let alone persuade them to join the Federation.
Given that, why would anyone ever bother going to other galaxies to meet new civilizations?