Oscilloscope measuring current that flows a few milliseconds
It sounds like you are inadvertently shorting out the resistor you hope to measure current through because the scope probes have a common earth point. If you have one scope probe's earth crocodile clip on the bottom of the lower resistor you can't arbitrarily connect the other probe's earth clip to a different point in the circuit because it shorts the two nodes together.

Take a step back and think what you are trying to do. If you connect one resistor (10 ohm) to your triboelectric generator and measure the voltage across this one resistor you can perfectly assume that the current through that resistor look exactly like the voltage across it.

This is basic ohms law i.e. I = V/R - if R is a constant then I is proportional to V and will be identical to it other than scale.

If it so happens that you see no voltage across the ten ohm resistor this is because your scope isn't sensitive enough. In which case, increase the resistor value or use a scope preamp.