The difference between the two sentences is that in the first, "population" behaves as an adjective that modifies "quality of life" (and "the" refers to "quality of life"), and in the second "the population's" is a possessive (and "the" refers to "population"). You can find the first usage with other words, like in "Increase the population density" (as opposed presumably to tree density, or housing density...). You won't typically find it with "quality of life" however, possibly because "quality of life" is always about people so you won't ever really need an adjective to modify it in the same way as more generic words like "density" or "average". I'm sure I could find contexts in which you would naturally say "Improve the population quality of life" if I tried, but I would have to try.
So both are grammatically correct, but "Improve the population's quality of life" is the natural phrasing you want to use.