Yes, you can.
But I guess it is more trouble than just replacing the PSU. You need to unplug all the PC case connectors (HDD & power LED's & power button, reset button, speaker, mic ) from the motherboard. And after your install the motherboard to the new case, you have to connect the HDD & power LED's & power button, reset button, speaker, mic of the new case to it. That is quite challenging if you are new to PC building.
But if you have a friend who can help you, a new case will open a lot of new upgrade options for you like bigger PSU, more powerful graphics card, etc...