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In Prisoner of Azkaban
"The falcon... my dear, you have a deadly enemy."
"But everyone knows that, " said Hermione in a loud whisper. Professor Trelawney stared at her.
"Well, they do," said Hermione. "Everybody knows about Harry and You-Know-Who."
Harry and Ron stared at her with a mixture of amazement and admiration. They had never heard Hermione speak to a teacher like that before.
But that was before we see Trelawney insult Hermione.
So why did Hermione dislike her from the start?
Well, partly, because as bleh says in their answer, Hermione and Professor Trelawney got off on the wrong foot rather. Right at the very start of the lesson, before the incident in the question, Hermione was affronted when Professor Trelawney pronounced that 'Books can take you only so far in this field.' This very much goes against Hermione's methods and inclinations.
'Nobody in my family's magic at all, it was ever such a surprise when I got my letter, but I was ever so pleased, of course, I mean, it's the very best school of witchcraft there is, I've heard - I've learnt all our set books off by heart, of course, I just hope it will be enough - I'm Hermione Granger, by the way, who are you?'
Harry Potter and the Philosopher's Stone - p.79 - Bloomsbury - Chapter 6, The Journey from Platform Nine and Three-Quarters
It's also possible that Professor McGonagall has already put Hermione off a bit:
'Anyway, who says the centaurs are right? It sounds like fortune-telling to me, and Professor McGonagall says that's a very imprecise branch of magic.'
Harry Potter and the Philosopher's Stone - p.190 - Bloomsbury - Chapter 15, The Forbidden Forest
But mostly, I think it's just that, throughout the series, Hermione is a sceptic and a rationalist.
For example, you may remember, in the Prisoner of Azkaban, Lavender's pet rabbit is killed by a fox. In the first lesson, Professor Trelawney had told Lavender that the thing she is dreading will happen on the sixteenth of October. On the sixteenth of October, Lavender gets the news and takes it as confirmation of Trelawney's prediction. Hermione is not impressed.
Hermione hesitated; then she said, 'You - you were dreading Binky being killed by a fox?'
'Well, not necessarily by a fox,' said Lavender, looking up at Hermione with streaming eyes, 'but I was obviously dreading him dying, wasn't I?'
'Oh,' said Hermione. She paused again. Then -
'Was Binky an old rabbit?'
'N-no!' sobbed Lavender. 'H-he was only a baby!'
Parvati tightened her arm around Lavender's shoulders.
'But then, why would you dread him dying?' said Hermione.
Parvati glared at her.
'Well, look at it logically,' said Hermione, turning to the rest of the group. 'I mean, Binky didn't even die today, did he, Lavender just got the news today -' Lavender wailed loudly '- and she can't have been dreading it, because it's come as a real shock -'
Harry Potter and the Prisoner of Azkaban - p.112 - Bloomsbury - Chapter 8, Flight of the Fat Lady
Again, in the conversation with Xenophilius Lovegood about the Hallows.
'But then ... do you mean ...' said Hermione slowly, and Harry could tell that she was trying to keep any trace of scepticism out of her voice, 'that you believe these objects - these Hallows - actually exist?'
Xenophilius raised his eyebrows again.
'Well, of course.'
'But,' said Hermione, and Harry could hear her restraint starting to crack, 'Mr Lovegood, how can you possibly believe -?'
'Luna has told me all about you, young lady,' said Xenophilius, 'you are, I gather, not unintelligent, but painfully limited. Narrow. Close-minded.'
[...]
'All right,' said Hermione, disconcerted. 'Say the Cloak existed ... what about the stone, Mr Lovegood? The thing you call the Resurrection Stone?'
'What of it?'
'Well, how can that be real?'
'Prove that it is not,' said Xenophilius.
Hermione looked outraged.
'But that's - I'm sorry, but that's completely ridiculous! How can I possibly prove it doesn't exist? Do you expect me to get hold of - of all the pebbles in the world, and test them? I mean, you could claim that anything's real if the only basis for believing in it is that nobody's proved it doesn't exist!'
'Yes, you could,' said Xenophilius. 'I am glad to see that you are opening your mind a little.'
Harry Potter and the Deathly Hallows - pp.333-4 - Bloomsbury - Chapter 21, The Tale of the Three Brothers
Indeed, so characteristic is this of Hermione, that Dumbledore relies on it. This is why he gave Hermione The Tales of Beedle the Bard.
'I am afraid I counted on Miss Granger to slow you up, Harry.'
Harry Potter and the Deathly Hallows - p.577 - Bloomsbury - Chapter 35, King's Cross
Myself, I'm very much with Hermione on this and I can say that, if I were in a classroom with someone like Trelawney, and I heard them talking like Trelawney does in the question, I would be similarly scornful and disdainful. Others wouldn't and, because I like rep points, I won't go into all of that here ... :P But it's certainly nothing to be surprised about.
The main issue there was that Trelawney was in fact a fraud who was bluffing her way through everything in class. Dumbledore admitted as much in a private conversation with Harry, telling him that he was keeping her on only because she once, many years ago, actually produced a real prophecy.
I did [hear the prophecy]," said Dumbledore. "On a cold, wet night sixteen years ago, in a room above the bar at the Hog's Head Inn. I had gone there to see an applicant for the post of Divination teacher, though it was against my inclination to allow the subject of Divination to continue at all. The applicant, however, was the great-great-granddaughter of a very famous, very gifted Seer, and I thought it common politeness to meet her. I was disappointed. It seemed to me that she had not a trace of the gift herself. I told her, courteously I hope, that I did not think she would be suitable for the post. I turned to leave. ..."
Which is when she channeled the prophecy about Harry (or Neville) taking out Voldemort. She was afterwards unaware she'd even done it.
A typical student would just let a BS'ing teacher slide, because a good grade is their ultimate goal. Some who are into that kind of thing might even embrace the mystic hoo-ha. But Hermione was the kind of person who cares more about learning and truth. A person getting away with spouting nonsense and being treated like a sage is never something she could put up with. Its just not in her.
And for Trelawney's part, for a person who exists on people accepting their BS, the one person in class who won't let it slide has to be really threatening. So she's not going to like Hermione one bit either.
She thought Divination as a subject was petty and chance.
In Prisoner of Azkaban
Books can take you only so far in this field...."
At these words, both Harry and Ron glanced, grinning, at Hermione, who looked startled at the news that books wouldn't be much help in this subject.
She had thought that Divination was stupid because it wasn't concrete knowledge.
Although I cannot recall a specific place in the book where this is spelled out, Hermione may also have learned to dislike fortune-telling during her pre-Hogwarts years. We all know that while there some people in the Muggle world who attribute efficacy to the various kinds of fortune-telling, the bulk of humanity regards the whole field as claptrap. Hermione likely brought this attitude with her to Hogwarts.
Seeing that the field was held in low esteem by the instructors she most respected could only have cemented that attitude.
