I've been looking at the Gates Carbon Drive and was interested to know if it would it be possible to put a Gates Carbon drive system on a Trek FX-S5 ? I'm planning to use the bike mainly for commuting to work. So want to avoid the Oil/Grease/Chain mess without getting into a Lycra suit.
No. Installing the belt requires a frame with chainstay or dropout that can be split so that the belt can be installed.
For avoiding dirtying your clothes, there are solutions like chain guards, reflector straps or plain old rolling up your pants or stuffing them into socks.
But if you're dead-keen to do this there may be a way to get a proper frame-builder to cut the stay and fit some sort of a "gate" or structural member that can be opened, then closed to restore rigidity. Something vaguely like the threadded latch on a climbing carabiner?
You'd also swap out both rear dropouts with horizontal ones for tensioning the belt.
Expect this work to cost a lot of money, and there will be repaint costs too.
If you have an aluminium frame this will basically be impossible. With steel it should be cheapest, and carbon fibre I don't really know.... Do talk to a qualified and expert framebuilder, don't try and do it yourself.
Your final option is sell off this bike and buy one with the specs you really want. A racey bike like your trek, with a carbon belt and an IGH or singlespeed is relatively uncommon.