If that happened enough times, over a long enough period, might monsters start breeding faster, in general, to keep their populations going
This question implies a teleological conception of evolution, where animals evolve to fulfill some sort of purpose. Neither animals, nor evolution, act with survival of the species as a goal; species moving towards higher fitness is simply an emergent property. Thinking of species "trying" to survive is, in some cases, a useful abstraction, but you need to avoid taking it literally.
Assuming normal nutritional needs (the monsters are not supernatural, for example), could this situation sustain an increase in the breeding of monsters and thus an increase in monsters?
You're asking two completely different questions, and treating them as the same. Increase in breeding and increase in monsters are completely different things. If the increase in breeding is in response to higher mortality rates, then you can have the first without the second.
As else being equal, evolution favors traits that increase breeding. So animals will already be "trying" (again, this is just a metaphor) to breed as much as they can. It's not like in a world without adventurers, predators will be "holding back" their breeding because "there's already enough". Whatever forces that limit their breeding without adventurers, will also limit their breeding with adventurers.
It is possible that excessive killing of predators will decrease their numbers to below carrying capacity, and there will therefore be an excess of food that supports increased breeding, but this will simply allow the predators to rebound to carrying capacity. It would not explain a sustained excess above carrying capacity.