No. Unless your scripts are saved in a cloud storage system like iCloud, DropBox or Google Drive and your computers are set to synch files via that system, your Final Draft scripts will not automatically be placed onto the other computer.
They must be copied to the new machine via USB stick, email, external hard drive or some other means.
Final Draft files work the same way as any other file and do not require special handling. However you get your other files onto the other computer is how you can transfer your FD files as well.