I saw the film "The Quiet Man" the first time in the early 1950s and I do not Remember Duke Hitting Maureen with a Stick. I also have it on D.V.D. and a Very old V.H.S. Tape and I did not see that on them. 
Below is a story told by Maureen about the making of "The Quiet Man" and she does not talk about the Stick and I think that she would have! :lol:
MAKING OF QUIET MAN
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Take it from someone who has viewed The Quiet Man 50 times, (at least...more or less...) the latest being this evening on TCM. There is NO scene in which John Wayne's character, Sean, hits Maureen O'Hara's character, Mary Kate, with any stick of any size or kind whatsoever. There are two scenes in which he is handed a stick to beat her. Up until quite recent times (maybe still) it was legal in Ireland for a man to beat his wife with a stick as long as the stick/switch/cane was no bigger around than his thumb. The first time Sean is handed a stick, it is by Mary Kate herself, to demonstrate her contrition after she angrily drove home from town in the cart and left him to walk. The base of the stick appeared larger than a man's thumb (after 50 or so viewings one gets to thinking about such things), while the ends were maybe around the size of a man's thumb, but there were two of them (forked stick). Sean touches the stick to Mary Kate's flank (to show audience members unfamiliar with the custom the purpose of the stick) and then tosses it in the fire. The second time, when Sean was dragging Mary Kate home from the train, the lady from the railroad station, (the only female in the crowd, at least the only one to speak, and so should have been on Mary Kate's side,) after kindly helping a man return Mary Kate's shoe, turns traitor and offers him "a good stick to beat the lovely lady." Again, this is not a straight stick but a forked one equipped to damage several places at once. Sean politely accepts the stick with, "Thanks," and doesn't discard it right away, but neither does he strike her with it.
The other physical "violence" from Sean is a swat on Mary Kate's behind just before they take the cart into town, perhaps ad libbed by Duke or directed by Ford without informing O'Hara, who looked shocked, and then gave a breathless little laugh so as not to ruin the take. She later said that Duke can't have known his own strength and how much the blow actually hurt. In the scene on the way home from the train, Mary Kate swings at Sean ineffectually and he turns and kicks her rear end. This was choreography carefully worked out between the actors which the director pretended to admire as wonderful improvisation. Those and a few times of Sean picking up Mary Kate bodily are the only instances which could be interpreted as violence. When he threw her it was on a bed and not anything hard and I believe that was the only time he even raised his voice anywhere in the film, other than jovially (very well thought out for a movie titled The Quiet Man--notice that even in his confrontations with Red Will his voice is quite low and measured and manages to convey serious menace and determination). The subtext here I think is that Mary Kate was "a fine healthy girl" who gave as good as she got and Sean didn't have to worry about her being a shrinking violet who would fade away on him. They would likely produce healthy and spirited children.