No. Generally speaking, an arpeggio is just a chordal harmony that’s “broken”, so the notes aren’t always played simultaneously. Imagine how Weird Fishes would sound if Ed just strummed on every eighth note for the whole song — it’d serve the same role, but it’d sound much flatter. Of course, there’s no hard rules, and plenty of melodies contain arpeggios. But even then, the individual notes of the arpeggio are not moving independently, so it wouldn’t be counterpoint. And the way the arpeggios blur together on Weird Fishes clearly gives them a harmonic role, over which Thom can add melody.
One could make a good case that Jonny’s upper arpeggios are in counterpoint to Ed’s lower arpeggios. In that case, the arpeggios still are not counterpoint on their own, but they form counterpoint in combination. But given that their role in the song is more textural, I wouldn’t recommend using that as an example of counterpoint.
Probably the most compelling example of counterpoint on Weird Fishes is Ed’s vocals. Though brief, they do form moments of counterpoint when Thom sustains a note at the end of a line, and Ed adds his motif above it.