Other than digging through source code, the only method is by using scripts and math. Quite easy if the game has good mouse registration (i.e. no acceleration or smoothing).
The approach is to first measure the 360 distance, then the whole field of view (i.e. moving what's on your left edge to the right edge). Alternatively half the distance (crosshair to edge) and doubling it.
So if the 360 distance is 8181 counts and crosshair movement from left to right is 2415 counts, this means your FOV is 2415/8181*360=106.27 degrees.
This method have a few caveats though. For 3rd person you need to measure against an object as far away as possible. Skybox if possible (if it's static). In terms of distance, at least 1-2 km (over 1 mile). Also some game have your first person POV slightly in front of the center of rotation, this may throw off any measurements as well.