Where the sun shines the brightest
It could be somewhere in the wide open, or it could be related to something a character says.

And the lines have gone silent
The 'lines' could refer to communication lines which have 'gone silent'. This could refer to fishing lines as well, no clue how 'gone silent' relates to that.

There I sit, peacefully
The answer is sitting somewhere peacefully, maybe near a water fountain.

Calm like a gentle stream
The answer is linked to water in some way, and is very likely calm.

All in all, I think the answer might relate to the former Galbadia Garden headmaster (I forget his name) who is, by the end of the game, sitting near Mayor Dobe's house in FH.