Yes... and no...
The Earth rotates (once per 24 hours) faster than the moon orbits (once every 27.3 days). From this perspective, it's the moon that's the stator and the Earth that's the rotor.
The Earth's magnetic field was measured as part of several Apollo missions to the moon. The article uses the unit of Gammas. One gamma = one nano-Tesla. So the Earth's field strength on the moon is approximately 31 micro-Teslas.
Here's the bad news...
The rotor in an average industrial generator turns at about 150,000 times per minute. Since the Moon orbits once a month (more or less) and the Earth rotates once a day (more or less), then we'll simplify this and say that the Moon is just standing still and that magnetic field rotates through the stator once a day. Now, this technically doesn't stop you. It just means you need at least 216,000,000X more windings to achieve the same result. Remember, that's a multiplier. The windings in an industrial generator can literally be miles and miles long... so we're talking hundreds of millions to billions of miles of copper windings. And it gets worse...
The magnetic field strength of industrial geneators is measured in Teslas. Not micro-Teslas. Technically, this also doesn't stop you. You just need at least 32,000X more coil.
Our final multiplier (multiplier...) is 216,000,000 * 32,000 ~= 7 trillion.
Trillions of miles of copper wire....
There probably isn't enough copper in the planet Earth to build the coils you'd need to do what you want. But I could be wrong about that. Still, a bit impractical.
Don't let that stop you!!!
The fun thing about science fiction is that the word "fiction" is involved. If you only stick with science-fact, all you're writing is a documentary. What do I suggest?
That you use a material, let's call it worldbuildingidium, that's a billion times more likely to move an electron than copper. Now you only need big mother-hubbard geneartors and you're not depleting Earth's supply of copper to build a second moon just to run one factory. Food for thought.