+1. Wouldn't need a large amount, not so much as it would bother the donor tank, you're basically just seeding the new filter. Adding substrate from an established tank also helps, most solid surfaces with an oxygen supply in a healthy tank will develop beneficial bacteria, not just filter media.
Might sound contradictory but a key to getting up and running quickly is not to be in a hurry. Stock gradually (not meaning one or two fish at a time, just don't load up with full grown fish right away) or start with smaller fish that will grow out to fill the tank, feed lightly to sparingly at first. I've used the same simple formula a number of times to start new tanks without a hitch-- seed the new filter with some mature media, add some substrate from a mature tank, don't fully stock the tank right away or start with smaller fish, feed lightly at first, give the new tank some time to get fully up to speed.