Generally not until the late transition metals. A common explanation for why relativistic effects occur at very high Z numbers is that when the nuclear core charge increases, so too does the average speed of orbiting electrons because the Coulombic force magnitude becomes large. At some point the electronic speed is sufficiently close to the speed of light that relativistic effects can no longer be ignored. Be aware that there's no hard and fast cut-off for when this happens. Relativistic effects are treated as a perturbation, and they could be applied even in hydrogen if you wanted to. The effects are not great enough to observe really until about period six, in elements like mercury and gold.