I am trying to put together a "simple" circuit that will control a relay/transistor when the voltage from a source changes from 12.5vdc to 13.5vdc. Was thinking zener, MOSFET, etc .. what would work best/be simple?
I "think" the problem I would have with a comparator is that I only have the ONE voltage to look at .. the one that changes from 12.5 to 13.5. I could see using one if I had two different voltages .. a steady 12.5 and the the one that varies to compare. But maybe I don't understand comparators well enough.
A comparitor is best to monitor voltage levels and trigger when voltages change, but for fine thresholds like you are suggesting, you really need a stable reference. Typically a zener diode connected in series with a resistor would suffice, but alternately you can use a low voltage regulator IC such as 7805 (I recommend the T03 case style) which would work marvelously with a 12v input.
Since others are giving advice, so far without management intervention, I would point out that the charging circuits of such things as self-starting generators are prone to horrid surges. These could easily fry a 7805, even in a big case style. Additional anti-surge protection would be required before the regulator.
Note that it is also much, much safer to use a low reference voltage, so that the battery input can be well attenuated to feed the comparator. Additional clamping may be needed part-way along the input voltage potential dividing chain.
All of this may not be easy, especially when just beginning to get to grips with the basics.