Drive Bistable Single Coil Relay with two MOSFETS
Try this:

When Q1's gate goes high, the R1 K1 node is pulled to ground, allowing charge to flow through R2 and K1's coil, latching the relay's contacts in one state.

After a time, when Q1's gate drive goes low, the relay will have been latched, and since there'll be be no potential difference across the relay's coil the contacts will be stable.

Then, when Q2's gate goes high, the R2 K1 node will be pulled to ground, through Q2, allowing charge to flow through R1 and K1's coil, latching the relay's contacts in the other bistable state.

After a time, when Q2's gate drive goes low, the relay will have been latched, and since there'll be be no potential difference across the relay's coil the contacts will be stable.

D1 and D2 are catch diodes which will shunt the relay's turnoff spikes to the 12 volt supply, clamping the spikes to about 13 volts.