Why does Madame Maxime forgive Hagrid?

Question

In Harry Potter and the Goblet of Fire, Hagrid seemed to cause Madame Maxime great anger. For a while, he was really upset with her while she was aloof and angry toward him.

So why did Madame Maxime forgive Hagrid? How was this all settled?

Answer 1

Toward the end of HP and the Goblet of Fire, Hagrid and Madame Maxime are called to Dumbledore's chambers and given the details of their mission to try to win the trust of the remaining giants. It seems that the prospect of their upcoming quest is sufficient to make her at least pleasant toward him:

‘Minerva,’ said Dumbledore, turning to Professor McGonagall, ‘I want to see Hagrid in my office as soon as possible. Also – if she will consent to come – Madame Maxime.’

then some weeks later...

‘Bin havin’ a cuppa with Olympe,’ Hagrid said, ‘she’s jus’ left.’ ‘Who?’ said Ron, curiously.
‘Madame Maxime, o’ course!’ said Hagrid.
‘You two made it up, have you?’ said Ron.
‘Dunno what yeh’re talkin’ about,’ said Hagrid airily, fetching more cups from the dresser

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As to the outcome of this, we learn in HP:OotP that large amounts of enforced proximity and shared hardships has worked their usual magic;

‘Madame Maxime went with you, then?’ Hermione interjected. ‘Yeah, tha’s righ’,’ said Hagrid, and a softened expression appeared on the few inches of face that were not obscured by beard or green steak. ‘Yeah, it was jus’ the pair of us. An’ I’ll tell yeh this, she’s not afraid of roughin’ it, Olympe. Yeh know, she’s a fine, well-dressed woman, an’ knowin’ where we was goin’ I wondered ’ow she’d feel abou’ clamberin’ over boulders an’ sleepin’ in caves an’ tha’, bu’ she never complained once.’

Note also that much of her grumpiness in the HP:GoF seems to centre around her unwillingness to admit to being a giantess. By HP:OotP it appears that she's come to admit (at least to Dumbledore and his close allies) that she has giant heritage.

Answer 2

I think that Maxime's reluctance to admitting she was a half giant was clearly due to racial discrimination. Hagrid and Maxime are perhaps the only two examples of gentle giants in the story. I don't think it was so much that she didn't think Hagrid wouldn't know, but rather the fact that he would say it and potentially reveal what is already obvious. Kind of the keeping up appearances paradigm.

Keep in mind, most people saw those two as big people rather than half-giants. You wouldn't know unless they told you. Keep in mind the kids when they saw Hagrid for the first time (Harry notwithstanding and who saw him first anyway) getting off the Hogwarts Express looked up at Hagrid amazed at his size, but never indicated any fear that he might not be entirely human. Same is a likely reaction to meeting Maxime for the first time.

When the world was in greater danger, and needed the two half-giants' help to ally with the giants or at least keep them from allying with Voldy, it was probably more important to do the right thing and accept her role rather than keep up appearances.