The problem just comes from the fact the manufacturer hasn't provided a Windows 10 driver yet, at least from Windows Update.
Here is how to try and fix the problem:
Open your Device Manager: your Bluetooth device should logically show with a yellow exclamation mark indicating a missing driver.
Well, this is not exactly a step. It's only to check if your device is actually detected... :p
Since the driver isn't available (yet) from Windows Update, go to Levovo's drivers download section and find your laptop.
If there's an available Windows 10 driver, download and install it.
If not, then search for a Windows 8.1 driver (that should exist).
Once downloaded, right click on the setup program and select "Properties". In the "Compatibility" tab, check the "run this program in compatibility mode for" box, then select "Windows 8" in the drop down menu below. Click "OK" then run the program.
Windows 10 and 8.1 drivers architectures are very close, if not completely identical most of the time. It should work OK.
Important note
Most laptops have a way to disable Bluetooth to save the battery.
This actually hides the hardware from Windows, making it disappear from the Device Manager, so if you can't find it there, then you've probably accidentally hit the function key that toggles it on or off.
Check on your keyboard top raw for a Bluetooth logo and hit the key while the Device Manager is displayed: if it's the case, then you should see it coming back. You may need to hold the blue "Fn" key to access this function, depending on whether you've turned the classical function keys on or not from the BIOS.
I've searched the Lenovo drivers section, and no driver is currently available for Windows 10, explaining why it has been removed (it's not available either from Windows Update, of course).
However, you'll find one for Windows 8.1 that should be compatible if you install it the way I explained above (pick the one matching your hardware variant):
Broadcom:
http://support.lenovo.com/us/en/products/Laptops-and-netbooks/Lenovo-G-Series-laptops/Lenovo-G580-Notebook/downloads/DS037359
LiteOn:
http://support.lenovo.com/us/en/products/Laptops-and-netbooks/Lenovo-G-Series-laptops/Lenovo-G580-Notebook/downloads/DS037674