In case you need to switch between various OSes (be it Windows and Linux, or just various Linux distros).
On Linux:
/var/lib/bluetooth/[MAC address of Bluetooth]/linkkeys
you can as well just copy whole /var/lib/bluetooth directory.
On Windows:
Inside the registry:
HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\services\BTHPORT\Parameters\Keys
Note that to read it you need to run regedit this way:
psexec -s -i regedit.exe
And for that you need to download PsExec from sysinternals.
More details at this stackexchange answer.