It is helpful to distinguish between the $\Lambda$CDM model itself and the underlying physical mechanisms that may drive the values of its parameters (such as the cosmological constant $\Lambda$) at different epochs in the universe's history. Even though some of the underlying physical mechanisms may depend on quantum theory (and may not be completely understood), the $\Lambda$CDM model itself is purely relativistic and does not use quantum theory.
To use an analogy from classical mechanics, we know that the macroscopic behaviour of two surfaces in contact is affected by the coefficient of friction between them. So we represent this by a parameter $\mu$ in our equations of motion. The equations of motion are purely classical, even though the underlying mechanism that determines the value of $\mu$ depends on the microscopic nature of the surfaces and the forces between them, and so depends ultimately on quantum mechanics.