No you can't connect it as USB Storage.
To try to clear this up once and for all, the LG G2 with stock firmware cannot be mounted as a removable drive like you would a USB stick (ie. it doesn't support USB mass storage mode (USM) and isn't assigned a drive letter). Instead it supports Media Transport Protocol (MTP) and will be seen as a "portable device" in Windows Explorer. For most needs, such as copying files to and from the phone, MTP mode will operated just like mass storage mode, that is you can drag and drop files to and from the device and explore the contents of folders on the device. But the phone storage WILL NOT show up as a standard removable drive.
This is not a phone-specific issue - Android officially removed support for USM with the release of Honeycomb.
Also, not supporting USM is an advantage for devices like phones because it allows the phone (and OS) developers to handle the available internal storage better and use more efficient memory formatting techniques (because they don't have to support a FAT partition to use with USM).
By default when you plug your G2 into a computer the phone will popup an option list asking what you want to do. You should have the following options: Charge the phone, MTP, Tethering, LG Software and PTP. If you select MTP your phone will appear in Windows Explorer as a device and you can manipulate the files on it.
If you don't get the setting pop up on your phone asking you how you want to connect, go to Settings > General > PC Connection. Check the box "Ask upon connection" to enable this. Touch "Select USB connection method" to manually open the connection type screen and set/change how you want to connect. The method you select will be remembered next (and each) time you connect to a PC.