I am providing 1.8 V as supply in a single supply voltage follower circuit. Now my concern is if I put negative voltage (Range -1.3 V to -500 mV) in the non-inverting pin, what will the output from the voltage follower circuit?
Typically for 0~-300mV nothing much will happen except the op-amp will be railed near ground at the output.
Below -300mV, especially -500~-600mV or more (at room temperature), significant current will start to flow in the internal isolation and protection junctions. The output may shoot up to the positive rail (called "phase reversal") or do nothing much, but at some point, with many op-amps, the current may get to the point where permanent damage takes place.
Adding series current-limiting resistance may prevent that, but it's a not typically a good thing to do. Other op-amps on the same chip may be affected, for example, or supply current and heating may increase.