The more you study Quantum Mechanics, Quantum Field Theory and the mathematical procedure of quantization, the more you understand that "quantumness" is orthogonal to relativity.
A simple way of seeing this intuition is as follows. Canonical Quantization is a procedure that applies to the symplectic geometry of the phase space. But what is phase space, really? The usual definition is, it's the space parametrized by the values of coordinates and momenta chosen at an arbitrary instance of time $t$. It looks like this makes a reference to "time" and so the phase space and quantization should ultimately depend on this single instance of time, which naively looks to be contradictory to the spirit of relativity.
However, there's a different, much more profound way of defining the phase space that happens to coincide with the definition above for almost all physical systems. The phase space is, no more and no less than, the space of solutions to the equations of motion. There's no reference to an instant of time in this definition, or in fact even a reference to time itself.
For a system described by an equation of motion that's second order in some parameter $t$ that we can call time, these two definitions coincide: you have two degrees of freedom per point for a second-order equation, these are your coordinate and momentum.
However you can see where I'm going with this. Quantization is a procedure that deforms geometry on the phase space, which can be defined in a time-independent way. You could conjecture that quantization itself can be carried out in a time-independent way and that relativistic systems can be quantized. That conjecture would be absolutely correct.
There are a plethora of examples. Quantum Field Theories are fully compatible with Special Relativity. Topological Quantum Field Theories have the same extended group of symmetries (diffeomorphisms) as General Relativity does, serving as toy examples proving that conceptually quantization can be applied to General Relativistic theories, if it weren't for technical, mathematical problems.