Geiger counter pulse detector - too many pulses from amplifier
I'd clamp the base of Q1 to +5 and to GND. And bias the Darlington to almost-on, using the 4.7Meg and 1Meg suggested by the editor.

To clamp, simply hang diodes (installed to be reverse-biased) from the base to +5 (thus the diode "bar" to +5v), and from the base to Ground (thus the diode "bar" to the base.

Try it with considerable high resistors. E.g. 4.7 MΩ to +5V and 1 MΩ to GND.