How do I convert a 3-axis quaternion to a 1-axis x-y axis plane?

The MPU6050 has no yaw reference, and without one, it is not possible to determine the value you want.

All derived yaw values will always be relative to the starting yaw value, and will always drift away from that origin due to rate gyro drift.

For your vehicle it would be much easier to use a tilt compensated compass, for which you need a magnetometer and accelerometer. If you go take that path, be sure to calibrate the magnetometer, after installation in the vehicle.