It was a most unlikely encounter. Particularly, for John Wayne, the actor. He was accustomed to portraying people of all descriptions, and was exposed to a wide variety of individuals from all over the world, but never anything like this. On the other hand, for James T. Kirk, it was hardly a memorable experience. 

To occur at all, there had to be certain coincidences. If, for example, John Wayne, portraying a crusty ranch hand, hadn't decided not to go to town with the rest of the crew for a long lunch break, he wouldn't have had the chance to be alone, with the opportunity to concentrate on his characterization, in the solitude of this Arizona desert. And he'd not have met Captain Kirk.  Although he wanted privacy now, John didn't really mind when two intruders appeared out of nowhere. It was kind of interesting. One moment he was the only person around, and suddenly, these strangers appeared, dressed in some sort of uniform-like costume, one of them with real funny ears. If they were stuntmen, it sure was a good trick, he thought.

"Howdy. How the heck did you do that?", Wayne drawled, in character.
"Hello, I'm Captain James Kirk, Commander of the spaceship USS Enterprise. We boldly go where no man has gone before. and this..."
"Is that right?", interrupted John, continuing in character, unruffled, and mildly curious about how these loonies got there. "Where did you hitch yer ship?"
"To whom do we have the pleasure of addressing?," Kirk continued.
"Don't know if it'll be a pleasure. Name's John Wayne."  
"Mister Wayne, how do you happen to be here, alone, apparently with no other living souls within miles? How did you get here?"
'Yer pretty nervy to be askin' me that, pardner. How come you fellers ain't with the rest of the crew, chowin' down?"
"The rest of our crew is aboard the ship, waiting to take us back aboard, to report on any wildlife we may have encountered. Do you consider yourself a human?"
"What kinda crack is that?"
"Standard data. We need your cooperation, Mister Wayne. Before you interrupted me, I was about to introduce you to Lieutenant Spock ot the planet Vulcan."
"I'm pleased to make your acquaintance, John Wayne," intoned Spock.
"Likewise. Now, how come I never saw you two around before? Are you extras?"
"It's a bit complicated for aliens to understand, my friend," replied Kirk.
"Who are you callin' an alien, Jimmy?"
"My name is 'Captain Kirk,' and all populations we discover are aliens, and that includes you, Mr. John Wayne."
"Now, hold on here, Jim, Don't go gettin' snippy. I discovered you fellers. And  you better get back on the tavern set. The shootin' is supposed to start again after lunch."
"Shooting?"
"Sure, that's what we're here for.."
"How many men in your crew, Mister Wayne?"
"Only about sixty, for this shoot. Why?"
"Attention Enterprise! Beam us up, Scotty!!"

"Doggone! How the devil do they do that trick?"