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All 84-89 300ZXs came with standard Nissan R200 differentials(R200 is a Nissan designation for one specific design which shares major components such as gears and bearings with each other) in varying ratios and types:
Z31 models after 4/87 are the only cars to have r200 clutch LSDs from the factory.
The clutch LSDs are a direct bolt in replacement to any 300ZX that came with an open differential, but you may have to swap your stock driveshaft flange onto your new differential depending on what year/model you are putting the LSD into. To fit the viscous LSD to a car that didn't come with one you only need the half-shafts for the viscous differential on turbos, and on non-turbos you also need the turbo companion flanges. If you're putting a LSD on an early car you'll also need to swap the early pinion flange onto the differential from your stock differential. These differentials are indestructable, change the oil every year or two and you'll never have a problem. Nissan Motorsports sells complete ready to install R200 clutch LSDs for approximately $700-$1100 with your choice of standard R200 ratios.
Phantom grip r200 installation procedure is here. Note: Phantom Grip sucks, splurge for a LSD.
To tell if you have a differential, lift both rear tires off the ground and turn 1 wheel. If the opposite wheel turns the same direction then you have a LSD. If not, you have an open diff. The transmission should be in neutral for this.