Can I produce small signal gate-source voltage by applying small signal source-drain current?
Yes you can.
When Vds changes, drain-gate capacitance Vdg communicates some current to the gate. Then you have a potential divider formed by Cdg, and the parallel combination of Cgs and the source impedance of the gate drive circuit.



simulate this circuit &ndash; Schematic created using CircuitLab
Note that this is an AC effect, not observable at DC. More specifically, it is a high frequency effect thanks to the capacitances, and most severe with high speed switching edges, where MOSFETs are commonly used, and increase with increasing voltage.
This can either slow down the switching times, via what is called the Miller effect, or even turn the device on unintentionally when something else modifies Vds at a fast enough rate.