I doubt it's termites. Termites would leave definitive trails, made up of their excrement as they gnaw away. The trails would be all over between the wood panels.
Timbor is the favored treatment for Powder Post Beetles, but you don't need 1.5 lbs to treat your desk.
The holes were more likely made by powder post beetles. This is about the time of year (depending on location Feb to June, thereabouts) that PPBs emerge from their larvae stage, emerge from the wood as adults and fly off. Tap on the wood to see if fine "dust" comes out. If the holes were made years ago, there still might be dust inside, yet no bugs.... kinna have to use your judgement about this. If there are still bugs in the wood, they'll likely emerge soon, so keep an eye out for evidence of new holes. Otherwise, short of buying a 1.5 lb package (enough to treat half a barn full of wood) of Timbor, you could drench the holes with mineral spirits (no harm to finish or wood).
The holes may be old, also, and the bugs long gone. How long have you owned the piece? Also, the bugs may have been in that particular piece of wood when the desk was made. If PPB had infested the wood after the desk was made, you'd likely have holes in other parts of the desk. PPBs home-in on their birth place and will come back to the same place they hatched from, bore new holes and lay their eggs.... and the cycle starts over. Since you say other parts of the desk have no holes, it is most likely there are no longer bugs in the wood, just remnant holes. If the desk was removed from where the bugs originally entered, then they don''t know where the desk was moved to... they'd have to find some other home-wood to lay eggs in. Does this make sense?
In all, from what you say, I doubt there is still active bugs in your desk.... unless you see further holes show up and fine dust spilling out. They don't bore through finished surfaces. They bore into raw wood.
Google PPB damage, signs of and/or life cycle. In Europe PPB are better known as woodworm.