The first thing to realize is that nobody has a working theory of quantum gravity, so nobody can really answer your question.
As timur has pointed out, quantization doesn't necessarily imply discretization. It also doesn't work the other way around: discretization doesn't imply quantization. You can certainly use finite difference approximations to solve the Einstein field equations, and people do indeed do this, but all this will give you is an approximation to classical (i.e., non-quantum-mechanical) GR. This is the kind of thing relativists do, for example, when they simulate violent classical processes like the mergers of black holes and the subsequent emission of gravitational waves.
Having said all that, there are a couple of leading candidates for a theory of quantum gravity, which may or may not be equivalent if you work out their detailed implications (which nobody has been able to do). These are string theory and loop quantum gravity (LQG). LQG does in some sense quantize spacetime, but you shouldn't take that too literal-mindedly. It's not distances that are quantized but areas and volumes. A naive way of seeing that you probably can't quantize distance by using some kind of grid is that under a Lorentz transformation, the grid spacing would undergo length contraction and time dilation. (Area and volume are preserved by a Lorentz transformation.) A theory similar to LQG is causal dynamical triangulation (CDT).
Scientific American has published a couple of popular-level articles about LQG and CDT:
Smolin, "Atoms of Space and Time," Scientific American, Jan 2004
Jerzy Jurkiewicz, Renate Loll and Jan Ambjorn, "Using Causality to Solve the Puzzle of Quantum Spacetime ," Scientific American, July 2008, https://www.scientificamerican.com/article.cfm?id=the-self-organizing-quantum-universe
Smolin has also written a nice popular-level book called Three Roads to Quantum Gravity, which is unfortunately getting to be out of date at this point.