Somewhat objectively speaking (and even then only in hindsight) the moment he had actually started to like the elf was when he realized that he did not have an attitude with Gimli because Gimli was a dwarf and he was an elf. Legolas was just like that, with everyone, and it was quite funny sometimes.
He just hadn't realized it yet because he went out of his way to not hear anything the elf had to say. But sometimes, he had no other choice. Like at night during mealtimes when they all sat together around the fire.
The hobbits were having some sort of debate with Gandalf about how he had thoroughly soured the Baggins family name because Hobbits all across Hobbiton claimed he was a corruption of rashness and foolishness. Gandalf, of course, insisted that his influence affected absolutely nothing and they were just using him as an excuse.
Aragorn, having apparently noticed something more cheerful about the already annoyingly merry creature narrowed his eyes at Legolas from across the fire, "What's so funny?"
"Just thinking about something Mith-" He paused for not even half a second, Gimli noticed he did that often, stopped halfway through a word to change it. "Gandalf said." He turned a wicked grin on the wizard, who narrowed his eyes and puffed on his pipe but otherwise said nothing.
"Are you going to tell us?" Pippin asked around a mouthful of food, "I would like a laugh if you have one to give."
"Of course, dear Pippin, but first I think some thought would make it all the better." Legolas placed what he dared to call a portion of food next to him on the ground and sat up straighter. Gimli had to admit, he was a bit intrigued.
"Let us go through the list of our companions here today and I will explain. Let's begin with the hobbits." At being mentioned the all looked at one another in appraisal, Sam blushing scarlet under the scrutiny. "We are all aware by now of Gandalf's friendship with our dear Bilbo Baggins, and as such as known Frodo for most of his life. And, if I do recall correctly, Gandalf only knew of Bilbo because of his aunt on the Tooks family, correct?" The hobbits nodded.
The elf looked a bit pleased with himself, "Good. And, since Merry and Pippin are cousins and Pippin is a Took it stands to reason that he had also known both of them for most if not all of their lives." While the hobbits nodded again Legolas briefly flashed another smile to the still silent wizard, "I know how he loves fireworks on birthdays. And I already know that Sam would always come to visit for the stories when he was young whenever Gandalf would visit."
He cast his eyes around the fellowship, looking for his next target. They landed on Aragorn. "You, of course, having grown up in Imlad-" He paused again- "Rivendell well were acquainted with our ancient troublemaker here since the moment the twins brought you home."
"Ancient?" Gandalf grumbled to nobody no particular but otherwise did not interrupt.
The other human was next, "And Boromir, I know that Gandalf went to Gondor in search of answers about his suspicions with the Ring, but I also know he visited before. Human lives are not that long - comparatively speaking - and so you and your brother would have also known him since childhood, yes?"
Gimli was starting to sense a pattern as Boromir nodded. "I personally have been listening to you and my father argue since before I could even understand what you were arguing about, and not only did you take Glion on quite the cross-country journey I know for a fact that every time you came to visit us you would also visit Dale and the mountain."
They all waited for him to finish.
"Here we all are, all eight of us attempting to sneak one of the most dangerous pieces of jewelry across the face of the earth, during arguably one of the most dangerous times in ages, which just happens to be exactly the most rash and foolish thing we could possibly do. And every single one of us has known him our entire lives. I just think its funny."
And he really did seem to think it quite funny, because as soon as he was done talking he burst into laughter. Pippin, having also thought this hilarious joined him a few moments later which forced the other hobbits to join in as well. And hobbit laughter was hard to resist, even for grumpy old wizards.
