Is it possible to have a plural of "A good night's sleep"?
Would "Some good nights' sleep" be correct?
I'm thinking specifically in the sentence: "I'm looking forward to a good night's sleep"
It would lose its idiomatic nature. The trope "a good [noun]" is used in many other expressions, and this particular version depends on that, not on the object "night's sleep".
Consider these parallel constructions:
I'd avoid some, since it would usually be heard as "some (good night's sleep)," i.e. some quality sleeping time in one night. To avoid tying myself in knots like that, I'd usually rephrase: