No, Legolas does not appear in the original text of the Hobbit, even though Thorin's company does go through Mirkwood and meet the Wood-Elves and their king.
In fact, the King isn't even named in the book, not to mention any sons he might have, and the description of the Wood-Elves is a far cry from that of the Elves in the Lord of the Rings - wild and dangerous and fond of wine.
Their king isn't even given a name - he's just the Elf King.
It's clear that he didn't really have the whole backstory of the Elves in mind when he wrote them -
The Hobbit wasn't even originally planned to be part of the Legendarium of the Lord of the Rings and the Silmarillion, it just merged into it later.
scifi.stackexchange.com/users/4516/Avner Shahar-Kashtan
He does not explicitly appear, as the other answers here note, but in not mentioning his name it is also not made explicit that Legolas is not in Mirkwood when Bilbo and the Dwarves make their way through.
In fact, as Peter Jackson noted by putting him into The Desolation of Smaug , his apparent age and forest of origin make it clear that he certainly could have been one of the elves the Hobbits ran into during their adventures (or misadventures) in the forest.
As Jimmy Shelter has pointed out, there is no familiar interaction between Gloin and Legolas at the Council of Elrond, but this could easily be chalked up to the old Dwarven-Elven feud (although, of course, the real-world reason is that Legolas had not been invented when The Hobbit was written).
What is revealed in the Tolkien Legendarium is: Legolas was the son of Thranduil Greenleaf, the Elvenking of Mirkwood (formerly Greenwood)
The Mirkwood Elves were becoming increasingly withdrawn and xenophobic in the Third Age, making it very likely that the son of the Elvenking would have been among their number when the Dwarves' company entered the Elvenking's halls Elves did not reach maturity until near their hundredth year, meaning that Legolas' adventures with the Fellowship must have happened after his first century of life.
Counting back to Bilbo's journey with the Dwarves, Legolas was (at the youngest) in his mid to late twenties during those events, and probably far older (the film producers claim he was 2,931 years old during the adventures of the Fellowship, but that has no basis in the book's canon.
More telling is, as Paul Griffiths notes, he says he "feel[s] young again" when venturing through Fangorn after the Fellowship fragments).
So clearly, it is possible (and maybe even likely) that the Dwarven Company met Legolas, though the book never mentions this meeting.
Tauriel, on the other hand, was invented entirely for the film adaptation.
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No, he was added for the movie.
They may have seen or even met him, but he was never mentioned in the books, let alone by name.
If you search
The Hobbit for Legolas (which is easily done on Google Books ), you will not find that word.
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Legolas is never specifically mentioned in the book.
However, he may very likely have been present in Mirkwood at the time that Thorin's company was imprisoned there.
Many of the answers here say that there was no familiar interaction between Gloin and Legolas at the Council of Elrond, but, respectfully disagreeing, there actually is in fact an instance of familiar interaction between Gloin and Legolas at the Council of Elrond.
Legolas explains to the council that Gollum has escaped from the elves in Mirkwood where he was being held because they were perhaps too kind to him by allowing him special privelages (specifically allowing him to go outside to climb a tree while under watch of an elvish guard.)
To this Gloin makes a reply:
"You were less tender to me," said Gloin with a flash of his eyes, as old memories were stirred of his imprisonment in the deep places of the Elven-king's halls.
"
(This is page 249 in my copy of The Fellowship.)
Whether or not "You" referred to Legolas specifically or to the Mirkwood Elves in general is unclear, but given the context in which Gloin replies, I believe that Gloin does in fact remember Legolas personally from his imprisonment with Thorin's company in Mirkwood.
scifi.stackexchange.com/users/51147/Ryan Luetters
No, Legolas does not make an appearance in the novel, though Thorin and company are captured by Wood-Elves.
None of the elves are named explicitly in the novel; even the king is simply referred to as "The King" or like variations.
Much of the movies deviate from the original story line.
For example, in An Unexpected Journey , the rock giants were added in simply for more action.
They never appear in the book.
Nor do the orcs; there are goblins, briefly, but they don't chase the dwarves and hobbit like in the movie.
scifi.stackexchange.com/users/17270/Dragona13
This subject causes alot of confusion between the bookies and the filmies.
Seeing as the books were written by Tolkien himself
we should use only the books for reference.
There is no mention of Legolas in the Hobbit.
The Elf Legolas Greenleaf is actually first mentioned in The Fall of Gondolin although we can't be too sure if this is actually the same Legolas.
When Legolas first realizes he is about to face Durin's Bane in Moria he clearly loses his cool.
While all elves hated balrogs if Legolas was at Gondolin he would have witnessed the deaths of Ecthelion and Glorfindel not to mention the general devastation wreaked by the balrogs.
This example of genealogy suggests that he is of the Gondolindrim but not if he is Noldorin.
Being the son of Thranduil means he would be Silvan.
After giving this great thought, I have come to the conclusion that the two elves are most definitely two separate elves.
The Legolas of Gondolin was of the House of The Tree and members of that house fought with heavy spiked clubs and slings as opposed to the marksman skills so noteworthy during the War of the Ring.
He lead the survivors out of the Ecoriath and was a known scout with excellent sight and hearing.
These skills I feel would be inherent to all elves rather than to only Legolas.
scifi.stackexchange.com/users/31828/Omar Devon Little
Legolas is the son of Thranduil, the King of the Elves of Northern Mirkwood. He is also a member of The Fellowship of the Ring in  The Lord of the Rings . Legolas also appeared in the films of  The Hobbit . Does he appear in the book  The Hobbit ?
Is Legolas in The Hobbit Book?
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<tolkiens-legendarium><the-hobbit>
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scifi.stackexchange.com/users/26703/Kevin The Knight
There is no mention of Legolas in the Hobbit.    Legolas does not appear anywhere in the novel the Hobbit.
Legolas does not appear anywhere in the novel the Hobbit.