Hermione found out that she was a witch when she received a Hogwarts letter, describing it as a surprise:
“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?”
— Philosopher's Stone, chapter 6, The Journey from Platform Nine and Three-Quarters
Of course, that doesn’t mean that Hermione didn’t exhibit any strange behaviour before she received her letter. JK Rowling described her parents as “a bit bemused” by her in 2004:
Does Hermione have any brothers or sisters?
No, she doesn’t. […] You see so much of Ron’s family so I thought that I would keep Hermione’s family, by contrast, quite ordinary. They are dentists, as you know. They are a bit bemused by their odd daughter but quite proud of her all the same.
– JK Rowling at the Edinburgh Book Festival, August 2004
It’s not clear whether they were bemused before or after learning she was a witch. Afterwards would be natural, but before would suggest she was exhibiting magic from a young age.
I don’t think canon addresses Hermione at a young age, or whether she shows signs of magic like Harry. (At least, not yet. Perhaps we'll find out more in a Pottermore update.)
ETA: Anthony Grist makes a good point in the comments that Hermione must have shown magical ability at some point, for her to be admitted to Hogwarts. On Pottermore, the process by which students are selected is described: it’s the Quill of Acceptance and the Book of Admittance (sample content; no account required), which contains this passage:
At the precise moment that a child first exhibits signs of magic, the Quill, which is believed to have been taken from an Augurey, floats up out of its inkpot and attempts to inscribe the name of that child upon the pages of the Book.
It also notes that “no child has ever been admitted whose name has not first been inscribed on the book’s yellowing pages”, so Hermione must have shown magic at some point, at least to the degree demanded by the Quill and Book.