UPDATE: I was able to measure activated paper alone without affecting detector by any other particles from the cell.
By repeated measurements I can now conclude that what I am measuring are really Alpha particles and likely nothing else.
There is another possible explanation that you should try to rule out. Geiger counters are sensitive to electrostatic fields. Here is a report of someone who discovered that the static charge on plastic bags was enough to increase the apparent background counts on their 44-9 Geiger-Müller (GM) pancake probe.
https://hps.org/publicinformation/ate/q10421.html
It is possible that your experiment is depositing charged particles on the paper and that is creating a high enough electrostatic field to increase the counts in your Geiger counter. This may be a more likely cause than alpha particles.
You could test this theory by trying to charge up a paper with a static charge and see if it produces similar counts. Or you could buy or borrow an electrostatic field meter of the type used in clean rooms and see if the paper from your experiment has a higher electrostatic charge after activation.