Just taking the filter over may not mean the bacteria will automatically grow in the new filter. Bacteria are hard to get a handle on, suffice to say there are enough bacteria in your tank to cope with the nitrogen cycle. But there's no way of telling where in the tank they are.
It's accepted that most of them are in the filter (it's dark and there's a flow of water, which means a flow of nutrient for the bacteria, plus the filter media - floss, sponges, ceramic rings etc - are designed to be a great home for the bacteria).
If you put your old filter in the new tank along with the new one, there will be a transfer of bacteria from one to the other, but bacteria only multiply to accommodate the amount of food available, so you won't automatically get double the bacteria just because you have two filters. there's a chance that the existing colony of bacteria in your existing filter will just continue to stay where they are.
The way to make sure your new tank gets a good colony of bacteria is to take the media (sponges, rings etc) from your old filter and put it in your new one. This can be done in several ways. If you have the same media in both filters you can do a swap from one to the other. It's pretty safe to take up to a quarter of the media from one filter to the other without issues, bacteria colonies can double in size well within 24 hours if the conditions are right.
If it were me and I was living in the UK where you can actually get hold of ammonia (not like Australia), then I'd be swapping some of the media from the old and new filters, setting up the new tank and performing a fishless cycle (adding ammonia to the tank and letting the bacteria establish before adding fish). If I were to do it myself now (without ammonia) I'd probably do the media swap. then slowly add fish from the old tank to the new and watch the water quality like a hawk.
If you have questions about what filter media you have and how to swap them, please let us know what you have in your filters and what you should be doing to transfer the bacteria. Sponges can be cut into pieces to fit in your new filter if necessary, for instance.
Best of luck with it all.