The Veela show up a few times, but the moments with Harry are pretty brief. The first big one is at the Quidditch World Cup in 'Goblet of Fire'. He sees them as part of the Bulgarian team's mascots. The description is wild – these beautiful, silvery women who turn into scary, bird-like creatures when they get angry. Harry's just watching from the stands with everyone else, caught up in the spectacle. It's more about establishing their magical nature than a real interaction.
Later, Fleur Delacour is introduced as part-Veela, and Harry meets her at the Triwizard Tournament. There's that scene where Ron is all dopey around her during the Yule Ball, but Harry seems less affected, maybe because of his mother's protection? It's never stated outright, but he notes her allure and finds her impressive yet annoying sometimes. The most direct interaction might be when he has to rescue her sister Gabrielle during the second task. Even then, it's frantic and task-focused, not a chat. So it's mostly observational stuff for Harry, which fits – he's often on the outside looking in at the weirder parts of the wizarding world.