According to the specs linked in your question, the lights will operate to as low as 10v. It's unlikely that your convenience outlet has low-voltage protection. This means that the lights will be able to drain your primary battery to as low as 10v before they cease to provide illumination.
Ten volts is far lower than considered safe for conventional starter batteries. With a rating of 42W, the lights may draw as much as 4A while being used, allowing for inefficiencies and wiring losses. It's safer to err on the pessimistic side for calculations of this nature.
You could perform a real-world test by using a clamp-on DC ammeter and observe the current draw with the lights on and off. That would give you a more certain basis for the math.
Examine your battery to determine the amp-hour capacity. For an example, I'll use the easy math of a 100 Ah battery. You don't want to drop below fifty percent discharge for a lead-acid battery, giving you fifty amp-hours with which to work.
The four amp draw in a fifty amp "fuel tank" means better than twelve hours use before you should be concerned.
The answer to your question is No, the truck won't turn off the lights. The real answer is Yes, you can excessively drain your battery.