Do carrier-less militaries maintain carrier qualifications on allied boats?
The problem with the carrier landing is currency and recency of experience


  Example of use would be in an emergency, i.e., land their F/A-18's on
  a US carrier even though they won't be able to takeoff, and the
  aircraft will likely need to be shipped.


The approach to landing on a carrier is a specific technique, wherein an angle-of-attack approach is flown all the way to the deck(and a very firm landing) with the intent of landing on the centerline and catching the 3-wire.  An acceptable approach catches any of the wires (there are 4).  The counter intuitive(for someone not trained in CV landings) advance-throttles-to-full- power-upon-landing ensures that if the wires are missed (Bolter!) one goes around and tries again if there's enough gas.


An approach to an airfield typically does not entail this, but instead includes a higher speed approach, and a flare which reduces the impact force on the aircraft (but of course requires a lot more runway).    

If the pilot of the F-18 has never had the initial qualification on a carrier(which takes hundreds of field landings and a Carrier Qualification flight), making that approach becomes a non-trivial exercise in risk assessment.  

If you aren't trained for it, why are you doing it?


Is the captain of the ship willing to risk a mistake that will damage dozens of aircraft on the deck and perhaps kill a few people?
Is the pilot willing to risk the approach that he or she is
untrained for, with the known lethality of failed attempts?  

The severity of failure can be seen in youtube videos of ramp strikes on carrier approaches.  I suppose the pilot could ask, but the Captain of the Ship and his Air Boss might refuse permission.  See risk analysis above.  During a shooting war ... who knows what people might try?
Here's an A-7 ramp strike that's a vivid example. 


To demonstrate the complexity of the matter:

The USN requires both an initial qualification and currency in order for its pilots, who land on carriers as a matter of standard operating procedures, to do a certain number of landings per quarter/per year/etc in order to remain current. If you aren't current you have to get recurrent to land on the Carrier.  

But what if that pilot had a PEP tour?

The US and Australia, among other navies, have a "Pilot Exchange Program" where we trade officers who fly in our services.  Part of building alliance relationship.  So, if this Australian pilot had done a PEP tour in USN hornets, and been a carrier qualified pilot, he may not be recent or current, but he knows the basics of how to fly a successful approach.  It's still a risk assessment issue since the Captain of the Ship, his Air Boss, and this pilot know that he isn't current, and that his kit may be out of configuration compared to USN hornets, so that he's flying a degraded or visual only approach.  

He can ask, but a decision to try it will take the safety of the pilot, his aircraft, and everyone/everything on that carrier's deck into consideration first.  (Low probability, but on a given day, who knows?)