Change Nether Quartz so that it changes to the environment that you are in. For example, Nether quart's background would be stone if in the overworld, and yellow if in the end. This may be used for building, as i believe it would look amazing
I severely dislike this idea. One of the best parts about Nether Quartz is it has a clean white texture that no other block has. The closest is White Concrete which is more gray.
Making Nether Quartz look like stone in the overworld would ruin the unique aesthetic niche it fills and make it look just like any of the dozen other grey blocks that exist. We already have Stone, Smooth Stone, Andesite (and the polished version), gray and dark gray wool and concrete, white Concrete (as I said it looks grey), and the powdered versions of those. We really don't need MORE gray.
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If you're talking about the ore, I dislike the idea as it is NETHER quartz, so it doesn't make sense for it to become embedded in stone when brought to the overworld. Not only would it be a strange feature for just this single ore to have and not all of them, but it would also make builds that recreate parts of the nether in the overworld impossible.
If you're actually just talking about the quartz block itself, I dislike the idea for the same reasons listed above.
Perhaps ore blocks could be "aggressive", in the sense that instead of taking on the properties of the surrounding world...the surrounding world takes on the propertiers of the ore (overworld ores would change netherrack to stone, whereas quartz would change things to netherrack.)
This would only make sense if quartz generated in the Overworld and/or it could replace stone (it can only replace Netherrack, much as Overworld ores can only replace stone, so either way there is no point to this) - or they added new "stone-type" blocks to the Nether or Overworld; for example, in TMCW I made it so that ores use metadata to determine the base texture so they blend in with underground biomes (stone is replaced with various other blocks):
Note that since these blocks use metadata to determine the base texture you can use Silk Touch to mine them and they will retain it; otherwise, they drop as the "default" stone so as to not clutter up your inventory, much as I made the "raw" variants of the 1.8 stone types drop as cobblestone unless Silk Touch is used - it would really be annoying otherwise unless the size of your inventory was increased.
That looks really amazing and in all seriousness I think it deserves enough support for Mojang to consider it. Perhaps the base material can be changed in a stonecutter or something by placing it and the desired base block together. But however that bit works, I think that quartz should be able to do the same thing, but it would only naturally generate as it currently looks. Overall I don't necessarily support how OP wants to achieve the end, but the end should definitely get support especially after seeing these pics. Well done!