Assuming by .EXE you mean a Linux ELF binary without additional dependencies, it would simply run on any Linux machine.
If you want to create an installer, you would need to package that binary as a .DEB and/or .RPM file, or a SNAP, or AppImage.
Otherwise you can simply compress it with into a TGZ file and will run anywhere you copy it.
I'd suggest that more to the point:
* By convention, the executable doesn't have an "extension" which is a Windows (nee DOS, CP/M etc.) thing.
* From the shell, you can run the binary by specifying its full path, or using e.g. ./MyProgram if it is in the current directory.
* Remember that unix filesystems are case-preserving and case-sensitive.
* You will find it more convenient to have a program on the path. By convention, locally-compiled programs go into /usr/local/bin
* The normal user will probably not have permission to write to /usr/local/bin, so will need to use sudo (qv) providing his own password for authentication.
* Files can be copied using the cp command. It might be necessary to use the -p option to preserve the executable mode (qqv).
* Installing a file into the computer's menu system depends on the GUI (desktop environment etc.) in use.
* There's a standard program "install" which can handle executable mode etc., but it won't solve the password issue or menu.
* If copying to a different machine (e.g. using sftp (qv)), you will need to ensure that any prerequisite libraries are present. This is particularly the case if one system is 32- and the other 64-bit.
* Stating the obvious, if you've used Lazarus to build a GUI-oriented program it won't run on a system which doesn't have a GUI.
MarkMLl