Can we check the specific gravity of a battery when its on charging?

It depends if the charger is well isolated or not, has nothing to do with the battery.
Power-up your charger with no load, output unconnected, and touch one of the wires. Depending where you stand, your shoes?you may feel some shock, when the battery is connected, there will be no difference in the shock level.
Be careful, batteries are allergic to sparks. Because when charging and discharging, they produce hydrogen that burns rapidly!
Turn the power ON and OFF away from the battery, then connect or disconnect the battery
Measure the gravity several minutes after the power was turned OFF, it is more reliable.

I hope this answers your question

First, voltage in most batteries is quite low. Not usually a shock hazard.
Second, hydrometer is usually made of glass with rubber bulb and tube. These are insulators.
Third, you can measure specific gravity when charging but this is not a good idea. Charging causes heat in the mixture and this affects the sp.g. Also bubbles of hydrogen are rising through the mix which will affect the reading.

Best to use the reading as an indicator. When you think the battery is charged, stop the charge and allow the electrolyte to settle and cool down. Then take another reading.