There appears to be an issue of Referential Opacity going on here. Which is of course rather apt, given your choice of subject. First off, though, let's get the technical point out of the way: you need to use predicate logic rather than purely propositional logic in order to make progress here. What you seem to be doing is using a constant term Quine and having it feature in two modal claims. Ideally, your two premises will look something like the following:
- □ is_Human(Quine).
- ◊ ¬ E x . x = Quine
In his "Three Grades of Modal Involvement", Willard V.O. Quine (I'll call the author Willard to avoid ambiguity with your named item Quine) discusses three ways in which we might discuss modal statements. The first is as an operator on sentences. An intuitive interpretation of your first premise using the first grade of modal involvement would be to say that the statement "Quine is Human" is necessarily true (or true in all possible worlds). The second way is as an operator on propositions, We would use the second grade of involvement to say that in all worlds, Quine is human. And the third is as a sub-propositional operator, which allows for modal operators for themselves to play a part in the composition of more complex propositions - e.g. there is someone called Quine, such that it is necessary that that person is human.
Now Willard thinks that the first analysis of necessity is okay and the second can be phrased in such a way as to reduce it to the first. But his problem with the third comes from the question of what object, if any, we should take as the reference of some term within the confines of the necessity operator. Necessity does strange things to our sense of how we ought to interpret descriptive singular terms; for example, while it is true that the number of planets in our solar system (8, for the sake of argument) is such that it is greater than 5, and while it is also necessary that the number 8 is greater than the number 5, it does not thereby follow that it is necessary that the number of planets in our solar system is greater than 5 (that it is necessary that there are at least 5 planets).
- □ 8 > 5
- number_of_planets = 8
- □ number_of_planets > 5 ???
The lesson Willard wants us to learn from this is that we can lose track of the reference of a particular individual term whenever we move it across a modal operator. So the problem with your argument, on this account, is that you are using a constant, Quine, to track your intended reference across possibilities, when in fact it is not at all clear what, if anything, you are tracking when doing so!
Now one way around this objection against certain kinds of modal technology is to move from thinking about Quine as a referring constant term (and hence involved in the third grade) to what it means to have the essential properties of being Quine - to rephrase your concerns in terms of a Trans-world Identification predicate (and the safe forms of the second grade). Let's assume that we have some property of what it takes for something to be Quine, and say that that thing Quineizes. Now your two premises take a very different form indeed!
- □ V x . Quineizes(x) -> is_Human(x)
- ◊ ¬ E x . Quineizes(x)
And these two coexist perfectly happily with one another, because the implication can still obtain even if nothing does in fact Quineize. This field of ideas is explored in the concept of Trans-world Identities - it too has quite a few problems, but if treated formally rigorously, it promises to avoid a lot of the sources of Willard's strictures against modal involvement.