Man, go for it!
Try to get Finale Notepad free trial (and if your lucky, a CD torrent).
I use Finale to write down stuff, and it's very easy to get the hang of.
If you want to compose without learning music, you might be out of luck, but if you learn enough to know the note names on the piano, and what a chord is, then you can use your imagination to make what you think sounds good. Your girlfriend will probably be extremely impressed that you even tried to teach yourself to write a piece on your own.
If you can get these down:
1- reading piano keyboard
2- reading music staff
3- understanding note values (quarter, half, etc)
4- learning major/minor tonalities
5- basic chords (major, minor, mixed)
Then you can probably come up with a pretty catchy pop-sounding thing. I know some people who have zero music background, but fiddle around long enough to make something decent sounding.
You don't need 3 years composition training, and classical lessons for x amount of time. Just look around online at some of the basic stuff I mentioned, and you'll have some sort of base to understanding what the music you hear LOOKS like on paper. Then go from there!