Recently, I've been making a 3D game (hint my questions), and I recently have modified a pixelization shader from a unity shader graph tutorial. The thing is, I want to render the shader on the WHOLE screen, not just a single object, but while researching, I've come across something called Framebuffer Objects (FBOs), which make it pretty much easier (basically just grabs the frame and turns it into an image for you to "process" it before it gets rendered on the screen), but I don't really want to use FBOs. How would I come across doing this? I've researched a little and came across using a quad, but since my game is 3D, not 2D, I have a perspective camera, so I don't know how that would turn out.
Are you using a game engine? If yes, your game engine likely comes in with a built in post-processing system/pipeline, in which case you shouldn't have to worry about frame buffers for simple post-processing.
If your engine doesn't have a post-fx system, or you're making an engine from scratch: yes you would use a quad. The key, is that instead of rendering/transforming the quad in world-space you would render and position it in view-space so it will be fixed in the screen instead of in the world. You'll do this changing whatever transformation matrix is being used to displace the quads vertices to a view-space-matrix.
You might want to familiarize yourself with the purpose of matrices (which effectively act as stores for transformations in 3D games) and the basic graphics / shader pipeline.
Here's a link which offers an overview of the role of matrices and spaces in realtime 3D rendering.