Seems your question is how the satisfiablity relation for a propositional modal language can be extended to a first-order modal language with quantifiers.
There are many options, none of them uncontroversial. Let a model of basic modal logic be a triple M = (W, R, V), where W is a non-empty set, R a binary relation on W and V a function from atoms to subsets of W. Satisfiability of a basic modal formula in a point from a model is defined in the usual way.
One option is to augment a model for basic modal logic with a non-empty set D and letting the quantifiers range over D. D is supposed to represent the possible objects, common to every world. So constant domain models are tuples M = (W, R, D, V), where (W, R, V) is a model for basic modal logic and D is a non-empty set. The satisfiability clause for the universal quantifier (x) is simple: M, w, g ⊨ (x)A iff M, w, g* ⊨ A, for every x-variant g* of g. (Read M, w, g ⊨ B as 'assignment g satisfies formula B in point w from model M'. As usual an x-variant of g is an assignment that coincides with g on all variables distinct from x).
This approach is simple yet not favoured by philosophers since it validates the Barcan and converse Barcan formulas.
Another option, probably due to Kripke, is to augment models of basic modal logic with domain functions d sending every world w to a non-empty set d(w). d(w) represents the objects that exist in w. So variable domain models are tupels M = (W, R, d, V), where (W, R, V) is a model for basic modal logic and d is a domain function on W. Now the truth of a quantified statement in a world w involves only objects from d(w): M, w, g ⊨ (x)A iff M, w, g* ⊨ A, for every x-variant g* of g with g*(x) a member of d(w).
This approach manages to invalidate the Barcan and converse Barcan formulas, but uses a certain free logic as its logical base yielding counterintuitive results as well. Constant-domain and Variable domain semantics can simulate each other and so are basically equiexpressive.
Besides these approaches there are many more such as counterpart-semantics or the admissible semantics recently formulated by Robert Goldblatt. If you are interested in these issues a good place to start is Fitting and Mendelsohn's book 'First-Order Modal Logic'.