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The current equation relating Vds ,Vgs and Vt is already known to us ,but if there is any way we can find out how it varies quadratically without using the formulae?

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    \$\begingroup\$ Are you asking for an intuitive approach from first principles physics that should tell you why? No equations, just a reason why you must find that kind of result? (For example, it is easy to show that angular momentum conservation must arrive from a simple consideration of the geometry of space.) \$\endgroup\$ – jonk May 25 at 18:07
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In an enhancement mode MOSFET, voltage applied to the gate terminal increases the conductivity of the device.

Logically, think of it in terms of the conduction gap being closed with applied gate -source voltage as a 1/r² effect as the electric field is increased so on a log scale it might look like a linear slope. enter image description here

Nanowire FET By Saumitra R Mehrotra & Gerhard Klimeck, modified by Zephyris - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=11299479

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