Will a Tamron 70-300mm f/4-5.6 DI LD Macro (Nikon AF) fit a Nikon 1 J1 camera?
With an adapter (The Nikon FT1), yes. As a native lens, no. (There are no third party lenses for the Nikon 1 system yet.)
Assuming you have the newer version of that lens with a built-in focus motor, it should autofocus and everything. It's my understanding that there is an older version of the same lens that relies on the camera body to have an AF motor; since the adapter doesn't have this it wouldn't provide AF in that case.
As we see from this earlier question, the current Nikon 1 lenses top out at 110mm, so getting to 300mm with an adapter may be compelling, particularly if you already have the lens. On the downside, the FT1 adapter is expensive, the lens probably will not be fast to focus, and the Tamron zoom is well over the weight limit where Nikon advises you to make sure the camera is supported to avoid damaging the mount.