No, this should not be done. The refrigerator must first be switched off for a few hours, so that the coolant can fall again. With older models it is advisable to let the refrigerator rest for about 12 hours before use. With more modern refrigerators, 3 hours is enough
An aquarium must always be moved empty. You can transport the fish separately in a cool box or a darkened bucket. Just like the fish, the plants must be transported underwater. For larger aquariums it is wise to use suction cups to lift the aquarium.