Why does electron velocity increase with the increase of atomic number?
Why does electron velocity increase with the increase of atomic
  number?


Newton's second law tells us that for an electron moving in a stable circular orbit around a nucleus the centripetal force pushing out must equal the electrostatic attraction pulling in, or

$${\frac{m_e~ v^2}{r}} = {\frac{Z~k_e ~e^2}{r^2}}~~(1)$$

where $m_e$ is the electron's mass, e is the electron charge, $k_e$ is Coulomb's constant and $Z$ is the atom's atomic number.  Rearranging equation (1) yields

$$v = \sqrt{ Z~k_e~ e^2 \over m_\mathrm{e} r}~~(2) $$

From equation (2) we can see that the electron velocity increases as the atomic number, $Z$, increases.

Edit: Response to OP's comment


  I am expecting some quantum mechanical explanation


As the electron velocity given by equation (2) becomes comparable to the speed of light, the mass of the electron increases due to relativistic effects.

$$m_{rel}=\frac{m_{e}}{\sqrt{1-(v_e/c)^2}}$$

As the electron mass increases, the s (and somewhat for p) orbital radius contracts.  So relativistic effects affect the electron mass and the orbital radius.  In terms of electron velocity, these 2 relativistic effects offset one another and the electron velocity is not affected; equation (2) is still valid.