No, all of these time frames you are thinking in are way too short. A good "biofilter" (the thing that defines a well-cycled tank) has to have two large colonies of two specific species of bacteria growing within biofilms that the colonies have created on the filter media surface. Both species are "autotrophic" bacteria and are very slow growers. Unless one starts somehow with colonies that have already grown, it always takes at least a month or so (no matter what the method) to get them up to the size needed for a tank to be ready to handle tropical fish in a healthy way.
A lot of what our beginners section has a good reputation for is helping beginners to have a hands-on experience learning about this whole "biofilter" thing so that their knowledge is flexible and helps them to practice good fishkeeping later on in the hobby when they meet unusual circumstances.
If you have a pond from which you could remove a smaller (or perhaps more than one) goldfish with the idea of putting it back in the pond later then yes, that would be a way to provide the bacteria with ammonia. The challange would be to get the ammonia concentration right. It can take quite a bit of experience to do fish-in cycling correctly, without harming the fish. If you take away the special case of the very experienced people who know how to do this, for most beginners fish-in cycling is just old school, replaced 3 decades ago by fishless cycling because it takes away the chance of permanently damaging the gills with ammonia or the fish nervous system with nitrite poisoning.
The trick with using goldfish would be to keep the ammonia and nitite(NO2) always below 0.25ppm. A small goldfish (or two, I just don't know) in a 55G might be just about right so that you wouldn't have to do large daily water changes to keep it below the 0.25ppm. But goldfish eat a large quantity of plant matter and are quite messy so its really hard for me to judge. Maybe another member will happen along with more feel for it.
If you have recently re-homed (in the last few days) all the fish in the 10g and no longer have any fish but still have the cycled filter, then the mature media from that filter could be put inside the new 55g filter to give it a jump-start on a fishless cycle that would be done using simple household ammonia. Putting the 10g on the 55g, even for a month, would probably only be getting it barely started and you would need an ammonia source.
The members here are great though, so keep discussing it and you'll no doubt come up with a good plan that suits you, does not harm your fish and also gets you ready for a great tank!
~~waterdrop~~
