It shouldn't, unless there's something on your skin like moisturizers or something organic getting into the tank, but typically no, it should not.
The API test kit is known to be a bit squirrelly, it sometimes reports a low level of ammonia when there's typically none.
If you've not yet added your fish, I'd encourage you to do a fish-less cycle using your Fritz Turbo Start and a source of ammonia that's not a living fish. I don't think it would take more than a few days if you're using the TurboStart, which seems to be the fastest acting of all these products by far.
You *can* add a fish right away if you're using the TurboStart specifically (but not Dr. Tim's One and Only or MicroBacter7, etc.), but it's largely frowned on to do it that way these days. Most feel that it's best for the fish to do a fish-less cycle, and then add your livestock.