Is the sentence "He quarrelled with me for a trifle "correct?

Yes , it is correct.
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There is no wrong in it. 
It means - He quarrelled with me for a small, inconsiderable thing. 

 

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Altough meaning wise correct, it sounds unpleasant. I would rather suggest, he quarelled with me for a small, unimportant thing.
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yes
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Yes
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Ni
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