Calculate the electrical current density (international metric systeml )
Is this result correct?

Not entirely.
Things like current or radius are &quot;physical quantities&quot;, i.e. they consist of a number multiplied with a unit. Any equation involving physical quantities are actually two equations: one of the numbers and one for the units. The product of  both equations needs to be correct at each step.
A simple example. You can write
$$ 1000 \mathrm{mA} = 1\mathrm{A}$$
That's perfectly correct since 1000 milliamps or 1 amp are the exact same physical quantity. However writing things like
$$I = 1$$, $$1000 = 1$$ or $$\mathrm{mA} = \mathrm{A}$$ are all wrong.
In your case, you should something like
$$J = \frac{I}{S} =  \frac{I}{\pi \cdot r^2} = \frac{1\mathrm{A}}{\pi\cdot 1^2 \cdot \mathrm{mm}^2} =  \frac{1\mathrm{A}}{\pi\cdot 10^{-6} \cdot \mathrm{m^2}}  \approx 318309.8 \frac{\mathrm{A}}{\mathrm{m^2}}$$
Every time you replace a physical quantity with a numerical value, you have to makes you use the correct unit as well.
In your last equation the first and the last terms are correct and they are current densities. However the middle terms are just numbers without units and that's wrong.