Hi wych2. The bacteria that utilize the waste from fish (drive the nitrogen cycle) attach themselves to the media in your filter, gravel, decorations, and pretty much everything in the tank. The media in the filter has a LOT of surface area for these bacteria to grow and that is where most of them grow (usually!). There are some in the water, but very little.
When cycling a tank, you are trying to grow bacteria. There are some bacteria on the fish you bring home and in the water they are in. But not enough to process the ammonia produced by the fish. The bacteria attach somewhere and multiply. This takes time but eventually, enough bacteria will grow to consume the ammonia and nitrites (waste) from the fish. Every time you add more fish, this process happens all over again.
So to answer your question, no. If you moved your filters and gravel over (along with new filters if needed for the bigger tank) then the answer would be yes. As MrGoodbytes has said, you can shorten the cycling process by using some gravel and/or filter media from the established tank.