I just looked at both photos one two different monitors (iMac 27" retina and NEC PA271) and do not see any grids in the areas specified or elsewhere in those photos.
This suggests that it may be something in your environment causeing the problem.
Are you viewing them at 1 to 1 or higher?
I don't see it either. Lots of jpeg compression artifacts and some noise in these images but that is probably from the software you circled the areas in and exported from. Do you see these grid patterns when you zoom in 1:1 or are they visible zoomed out? Your description sounds a bit like maze artifacts which I haven't seen in Lightroom for years and you should only see at 1:1 zooms. The images above look like they were looked at zoomed out suggesting the artifacts are visible to you when zoomed out. What camera are you shooting with? Any chance you could share a raw image that shows the artifacts through a dropbox link or some such?
The two JPG images in your first post have JPG compression artifacts in them. I have severely sharpened and enlarged by 400% the top right corner of the image with the taller rocks to illustrate these 8x8 pixel squares a JPG is divided up into, and highlighted one with a red square. The more compression (lower quality) used to create the JPG the more distinct these 8x8 squares are. The forum-upload process may be additionally compressing the JPGs so it's hard to tell what you're wanting us to see.
Are these 8x8-pixel squares that are caused by JPG compression artifacts what you're taking about, or are they merely obscuring what you're talking about?
If these aren't what you're referring to, then instead of JPGs, you need to be attaching PNGs of screenshots of the LR Develop Loup area zoomed to more than 1:1 or 100%. 4:1 or 400% would probably be sufficient. PNGs are needed because they will have zero additional artifacts, and screenshots are needed because otherwise you'll showing us compressed JPGs from LR which will have artifacts.
If what you're wanting to show us are in fact JPG artifacts from LR exports, then export such a JPG, and reimport it into LR, and show us a PNG of a screenshot of a 4:1 or 400% zoom area in LR Develop where using a PNG of a screenshot will let us see the actual look of the exported jpg without additional artifacts caused by the forum upload. Also please tell us what the Compression/Quality setting you're using in Export.
It's important to use LR Develop for these screenshots, because LR Library is showing previews which are JPG-compressed and will have these artifacts.
Yes! I have the same problem!!! And I was looking at this person's shot on 500px and he has the same issue. Great Sand Dunes in B&W by Jack Mclane - Photo 157159779 - 500px - have a look at the lower left corner of the picture - it seems to be a particular problem when there is significant small detail in the photo.
Can you show us an annotated screenshot of your own picture where you indicate the size and placement of the grid pattern you're seeing and also supply us an exported copy of the original raw as a DNG with the LR-settings embedded in it so others of us can try the image on our own systems and see if the same thing occurs? It could be an artifact of the browser displaying the image or something the GPU is doing wrong, or even some moiré pattern due to the lens corrections.