It does make sense to try and compile your app on Linux, at least for the first time because you will need to check that it works as expected. And you might find you have used some Windows only specific things that you need to fix. So, assume you need Lazarus installed on Linux system. Please read the relevant wiki page.
To transfer your file, ssh is great on Linux, (relevantly) hard work on windows. Lots of other options.
Are your Windows and Linux boxes separate boxes or are you dual booting ?
First, make a zip of the source. In Lazarus, on your windows box, choose Project-> publish. Tht will make a zip file very easy to transfer to a linux box.
If you are dual booting, then boot Linux, it will mount your Windows file system, easy to find that zip file that way.
Else, as Trev said, smb is a good option, make a 'share' in windows, put the zip file there. On the Linux box, most file managers include what ever is necessary to mount a windows SMB share.
Or move the zip file to google drive, drop box or what ever ....
Davo