You'll have a hard time finding a single, integrated boost converter capable of going from ~3.7V input to 500V, and even harder time finding one which has very low quiescent current...

(You said "lithium cell" and said it was rechargeable, so I assumed a Li-Ion/LiPo cell of some sort here...)

But let's start with simple maths first: assuming the output current is 10µA and you can find/design a converter that is 80% efficient (which IMO would be already pretty optimistic here at this output current), that would yield an input current of 10e-6*500/3.7/0.8 ~ 1.7mA. With a 135mAh battery, you'd get ~79h (a little over 3 days) of continuous operation assuming that nothing else is drawing current (which I doubt if your device is supposed to be logging data)... would that be what you'd call "extended times"? Or would you actually have your device log radiation only a fraction of the time, thus having a low duty cycle? This preliminary remark just to make sure your expectations are realistic.

One approach could be to use an efficient, low-power boost converter to go from 3.7V to an higher voltage and then cascading a Cockcroft–Walton multiplier (see:

https://en.wikipedia.org/wiki/Cockcroft%E2%80%93Walton_generator , and I think Dave also made a video about it) to go up to 500V. IMO, a Cockcroft–Walton multiplier alone to go from 3.7V to 500V would be bulky and never be efficient enough to get you the level of current draw you're targetting.

One possible such boost converter could be the LT8494:

https://www.analog.com/en/products/lt8494.html(typ. 7µA quiescent current!), set the output at the max. supported: 70V, then cascade a Cockcroft–Walton multiplier for a ~7x multiplication.

Another option would be to use a flyback converter, but this would require a transformer (don't know if it would fit your size requirements) and good luck designing one to draw that little current...