After the war, Harry finds an injured bird, maybe a beautiful gyrfalcon, and takes it home with him, intending to nurse it back to health. Perhaps he is doing it as a way for him to cope with the loss of Hedwig.
Fleur, single again, it turns out, has a knack for taking care of birds, and often helps him out, so the two spend much time together. Their shared worry about the bird, their shared efforts, it brings them closer together, until they like spending time with each other even without the constant worry about their patient.
Unknown to Harry, at least at first, the injured bird is in fact a Veela, perhaps Fleur's relative, who had been attacked, and fled after transforming into a bird.
Due to her injuries, she was too weak to change back.
Fleur was looking for the other Veela, trying to make sure she was fine.