What would be the atomic number of the atom whose 1s electron moves nearly at the speed of light? What would be the atomic number of the atom (may be hypothetical) whose $1s$ electron moves at $0.99c$ (the speed of light)?
Quantum mechanics might have an answer, but I do not know the necessary maths to calculate. I am interested in the answer.
In this article they say that the speed of the electron defines gold's property through relativistic quantum mechanics.
 A: You can get a back of the envelope notion of the energy of a inner-most orbital by just treating the problem as a hydrogen-like atom (not entirely fair and almost certainly a slight over-estimate but at least it is easy). You get
\begin{align*}
E_{1s} \approx \mathrm{Ry} * Z^2 = (13.7\,\mathrm{eV}) * Z^2 \;.
\end{align*}
Where $Z$ is the atomic number of the atom in question and $\mathrm{Ry} = 13.6 \,\mathrm{eV}$ is the Rydberg constant.
Then we can pretend this is kinetic energy and compute some kind of speed on that basis. (This is simpler but less exact than the computation suggested by Ruslan in the comments. Nor does it really mean that there are little ball-like object in there whizzing around along classical paths.)
If you are asking for a speed of $\beta = 0.99$ ($\gamma = 7.1$) then you are suggesting an kinetic energy of about $T = (\gamma - 1) m_e c^2 = 6.1 (5.11 \times 10^5\,\mathrm{eV}) = 3.1 \times 10^6\,\mathrm{eV}$.
Which suggests:
\begin{align*}
Z^2 
&= \frac{(\gamma -1) m_e c^2}{\mathrm{Ry}}\\
&\approx \frac{6.1 (5.2 \times 10^5 \,\mathrm{eV})}{13.7\,\mathrm{eV}} \\
&= 2.3 \times 10^5 \\
Z &\approx 480 \;,
\end{align*} 
give or take a small factor.
Even for $\beta = 0.9$ ($\gamma = 2.3$) you get $Z \approx 220$.


*

*For $\beta = 0.75$ ($\gamma = 1.5$) I find $Z \approx 140$.

*For $\beta = 0.65$ ($\gamma = 1.3$) I find $Z \approx 110$.

*For $\beta = 0.55$ ($\gamma = 1.2$) I find $Z \approx 86$.


All of these values are thoroughly relativistic, but as you can see the ultra relativistic regime requires unreasonable heavy nuclei.
A: Gold has a strong absorption line at 200-300 nm which is for blue photons. The complement of that blue is yellow, so the reflected light looks yellow to the eye. 
Yes, the effect is due to special relativity, but it is slightly different for different elements. The line for gold is the 5d to 6s transition, and the relativistic effect makes it come out to be blue. It is similar but not as strong for cesium, which also looks yellowish, just not as much as gold. 
Note that you CANNOT do a simple calculation to predict which elements will look yellow. If one thinks that something close to gold, at Z=79, will also look yellow, note that lead at Z=82 is not yellow at all. Note also that it is typically not the 1s electron's energy, it's really the transitions from other orbitals, and the orbital energy differences, that determine the light absorbed, and thus the color.
But most or all those heavy elements do have relativistic electrons. 
See it at https://en.m.wikipedia.org/wiki/Relativistic_quantum_chemistry, and also at https://jameskennedymonash.wordpress.com/2014/07/13/why-is-gold-yellow-the-chemistry-of-gold/
