Hi Benny. Nice to meet you. Well, it all depends on whether or not the filter sponges were kept moist and the tank was set up while the filter bacteria were still viable. If not, then you will have to generate a filter bacteria source in the 120L tank and the easiest way to do this is to put some of the filter material in the 60L filter.
This is how I cycled my second tank. I kept the new filter sponges in the old tank for a couple of weeks beforehand to "seed" with bacteria before I put them into the new tank. It might be worth doing that if you have the room in the filter.
Alternatively, after about two weeks, you could always divide the sponges between the two tanks and have half new (120L), half old (60L) in each. You could then slowly stock the 120L tank i.e. no more than 1-2 fish a week.
It is, in essence cycling with fish but the fact you have access to established filter material means that it won't take as long as a conventional cycle. Keep testing the water and do water changes if there are any traces of ammonia or nitrite and you should be ok.