Can I use the an FM transmitter that works with the iPod Nano 1st Gen with my iPod Touch 2nd Gen?

No, they are not all universal. Oftentimes different gens or models have different port placement, I recall I couldn’t use a Nano FM transmitter on my other iPods. The only guarantee that it’ll work is made by you physically connecting and testing it.

@dynamicduo Is it possible that it could damage it?

Well in some cases it won’t physically fit because some FM tuners for the Nano included a plug into the remote control port, so yes trying to shove that into an iPod Touch will obviously damage the part. I don’t know regarding voltages. It really depends on what type of an FM tuner it is. It could possibly work fine, it could not. So do you have a link to the FM tuner you have?