There's an ongoing disagreement bewteen Gimli and Éomer about Galadriel, one that is only resolved because Gimli accepted that Arwen was an acceptable alternative opinion.
From The Return of the King, chapter "Many Partings":
'Well, lord,' said Gimli, 'and what say you now?'
'Alas!' said Éomer. 'I will not say that she is the fairest lady that lives.'
'Then I must go for my axe,' said Gimli.
'But first I will plead this excuse,' said Éomer. 'Had I seen her in other company, I would have said all that you could wish. But now I will put Queen Arwen Evenstar first, and I am ready to do battle on my own part with any who deny me. Shall I call for my sword?'
Then Gimli bowed low. 'Nay, you are excused for my part, lord,' he said. 'You have chosen the Evening; but my love is given to the Morning. And my heart forebodes that soon it will pass away for ever.'
Would they have really tried to kill each other? Gimli killing the King of Rohan, I'm sure that would have gone over well with the people of Rohan. Or would this have been more like a non-fatal duel or something?