A new Arduino IDE for Galileo has been released recently. This IDE is the 1.6.0 and it doesn't show the option to update the firmware version. A new tool is used now to update the firmware version instead of using the IDE. You can download this new tool "Intel® Galileo Firmware Updater" from here: Intel® Download Center. You will find a useful User Guide here: Intel Galileo Firmware Updater User Guide 1.0.4
The version 1.5.3 is still available here: Intel® Galileo Boards and Kits — Software Downloads but I recommend you to use the newest version.