Wow, you’ve really got me stumped! When I first saw it, I wondered if it was some sort of moire, or something like that. The best example of it that I could find is here. The second image you posted looks somewhat similar to that pattern.
I guess my next question is, what was your “white” background made out of? Was it a painted wall, or a whiteboard, or some sort of material?
I did quite a bit of searching online, but found very little that dealt with the issue. I did find a few examples that look very similar to yours, but no one had an answer as to what caused it, or how to fix it. Here are two pictures on flickr that exhibit the same phenomenon, but they were both taken by the same person with an older Canon DSLR. That proves that it’s not just an isolated issue with your camera in particular, or Nikon DSLRs in general.
I’m guessing that as you said, it might have to do with sharpening in Lightroom, and someone else evidently had a similar opinion.
I also found a thread on Adobe forums where someone had a somewhat similar issue, but again, there wasn’t really any answer. And here’s one more example involving Photoshop as well.
One common theme seems to be that Lightroom is always involved…
Anyway, I would try a few things to see if you can clear up your trouble. First, try exporting to a .tif file, and see if you get the same issue.
Definitely try turning off all sharpening in Lightroom and export, then add some noise reduction and export, and so on, see if you can make it disappear. Try ‘masking’ the sharpening and see if it makes any difference. In fact, you could even try Lightroom’s built in moire tool. Make sure that you don’t have Lightroom set to resize/resample on export, and make sure that it also isn’t sharpening on export.
Let me know how it turns out, I hope it’s not too difficult to fix!