Just adding a new filter or filter media does not do a thing to create more bacteria. More ammonia does this. But, you should get some bacteria to colonize the new foam.
Most of the bactreria lives inside a biofilm it creates for that purpose. The film attached to hard surfaces. But a small percent of the bacteria will be motile. the are not that way to process ammonia in the water, but to be able to move to someplace else if the ammonia dries up. Of course in a tank there is no place to move like there may be in the wild. How many individual in a colony may become motile depents on the Nirtogen availability in that water, When this drops they set out for better places to live. This is one of the strategies the nitrifiers use to survive over time when bad conditions prevail.
What happens in a tank where a new filter is added is some of these bacteria which are motile will end up in the new media. But since the ammonia level is not changed the total number of bacteria in the tank will not change. However, ever day in a tank where the ammonia creation is stable, some of the bacteria will die and an almost equal number will be created by reproduction (the bacteria divide).
If the ammonia loading in the tank increase, the birth rate will exceed the death rate. However, if the ammonia level goes down, then more individual bacteria will than are being created by reproduction.
The result of all this is for any amount of bacteria to colonize the new media means an equal amount will decline in other media.
There are better methods for what you want to do. These all involve adding viable bottle bacteria or filter squeezing into a new tank and then dosing it with ammonia to make the bit of bacteria with which you start, reproduce. With the bottled bacteria you can put in as much as you need if you can pay for it. the mire you have the less time it will be before the tank is very ready for fish, lots of them. If you want to go this route I suggest Dr. Tim's One and Only or Tetras Safe Star or Safe Start Plus.
If you use filter squaeezings and only have a sihgle tank from where to get them, you will need to do an abbreviated fishless cycle to get the bacteria to multiply. If you can take squeezing from several tanks it will take less time to get the new media/tank cycled.
edited to fix typos