Induction power source across 1/2" of wood?
It is possible, but the energy losses are great, and the insulator (your desk) will heat up, probably causing heat damage.  Even then, the receiving coil wouldn't be integrated into the base of the lamp.

In short, there's no way this wouldn't be even more obtrusive than a cord hanging off the side of your desk.  If the desk is against a wall, probably better to put in a desk height electrical outlet.

The reason induction charging works for cell phones is due to tiny air gaps, low power requirements, and generally good heat dissipation compared to energy losses.  Only a tiny fraction of the used power actually makes it to the cell phone.  But the phone is storing that, so eventually (over the course of an evening) the phone is charged.  By the way, most phones lack receiving coils, so you still have to plug in a receiving coil and wire into the phone charging jack.

Power that is not directed tends to dissipate according to the inverse square law.  Even if it didn't, the losses in transforming the power to wireless and back are quite large.  Some of the wireless community is quick to point out the wire transmission losses, but that's nothing compared to wireless.

With a lamp, you'll need a lot more power, and continuous power, so the issues with induction fed systems will become a lot more apparent.  I would recommend a battery powered LED desk lamp (with a set of rechargeable batteries).  The LEDs will extend the time between recharges.