Question

Does Misty ever find out that Togepi knows Metronome?

Answer 1

She must not know Togepi can use Metronome because when Togepi uses Teleport in the anime so that the group can escape the charging Rhydon the group are confused on which of their Pokemon used Teleport.

Misty: It can't of been Togepi.

Then Togepi uses Metronome again in the same episode for the last time during the anime. And with Misty not even clocking on to a Pokemon using Barrier she still remains clueless.
Whilst on the airship it looks like she understands yet unfortunately she doesn't mention Metronome even though she believes Togepi saved them.

So I'm pretty sure that Misty is unaware of Togepi's Metronome but she does believe that it has moves.

Sources:

• Bulbapedia
• Youtube

Comments

It COULDNT HAVE been Togepi.

Fix yer grammar, nub.

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Ah I was too slow.

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"Do your thing, Experts"

Answer 2

One of Togepi's move is Metronome, which randomly selects a Pokemon move or attack. The anime used this move quite convenient to get the heroes out of sticky situations (though they never realize that Togepi was the one doing it)

So no, not even Misty ever realizes this at all. Weird how she didn't even bother knowing the moves Togepi had.

Source (The Answer)
Hope I helped!