Ok, so, versions first just so we're all on the same page.

Minecraft : 1.12.2
Forge : 14.23.5.2847
Quark: R1.6-178
Tinker's Construct : 1-12-2.13.0.183

Both Quark and Tinker's have been updated to the latest 1.12.2 version. So... After I created my world, created the TCon tables (Stencil, Part Builder, Tool Station) I noticed a ghostly image that looks like a trash can in the upper right two inventory slots of the Tool Station. No other GUI, just the Tool Station. After some research ( and possibly annoying KnightMiner over at the Tinker's Curse Forge page) and doing the dreaded 'uninstall a mod mid play though' it turns out to be Quark that's causing it.

With Quark installed

Without Quark installed

So, I don't know, there are no newer versions of Quark available for 1.12.2, I can't upgrade to fix this, not without changing Minecraft versions so... How do I fix this? Thank you for your time.

We're not updating 1.12.2 with minor bugfixes, unfortunately. You can, however, disable that individual feature from ingame by going to the q button in your title screen. (If not available, go to Mods > Quark > Config). The feature is part of Management.