No. When the Arduino output goes high, the base terminal is at 5V. This means the emitter terminal is about 0.7V lower, or 4.3V, and that's not high enough to turn on a non-logic-level MOSFET.
I'd suggest a circuit something like this:
Instead of powering an LCD backlight, you can turn on a MOSFET gate. Replace +5V with your +12V from the car battery on the right side, get rid of C1, and it is a reasonable beginning.
However....note that a car is a "hostile" electrical environment. If you just have a car battery in isolation it's OK, but if it's hooked up to a car you might see some nasty stuff, like temporary high-voltage spikes. You will have to condition/protect the circuitry from that (but let's not get ahead of ourselves).
The Rugged Motor Driver: two H-bridges, more power than an L298, fully protected