Yes and no. With all the filters you'll probably be ok from a cycle perspective. But you can't just buy fish, and add them to your tank without quarantining them first. You're taking a huge risk.
I'd leave one filter on the old tank, and quarantine the new fish there. Then after the 3 month quarantine, you can move them to the new tank.
The correct way, is to dismantle the old tank and filter. Sterilize them. Then fishless cycle with ammonia, then quarantine the new fish for 3 months, then add them to the new tank.