To download or export an file from Final Cut Pro (FCP), you must adjust the export settings from the default "QuickTime" (MOV) format to the "Computer" format. There is no separate "MP4 downloader" for Final Cut Pro; the functionality is built directly into the software's sharing options. How to Export Project as MP4

Follow these steps within Final Cut Pro to save your video as an MP4 file:

How can I export videos as .mp4 instead of .mov? : r/finalcutpro