Good afternoon, and thank you for posting on the Final Fantasy XIV Technical Support Forums.

I believe the game will only let you play it on one monitor, so using the fullscreen option will result in the second monitor being inaccessible. As stated in a previous comment, you can try the windowed or borderless fullscreen option (both of which can be found in the graphics/display settings in the in-game configuration settings). Either of these options should allow you to keep your second monitor open.

Please let us know if your issue gets resolved and what resolved it, so we can further assist other players. We hope we can be of assistance in helping you enjoy the exciting world of Final Fantasy XIV!