Your question seems to be very confused about terminology. So let's clear some things up.
There is a variant called $\sf ZFA$, or $\sf ZF$ with the existence of "atoms" or "urelements", which are objects that are not sets. In this context we have a predicate for "set" and a predicate for "atom", and we write the axioms of $\sf ZF$ for sets, and atoms are just objects which can be elements, but have no elements of their own (the empty set is a unique exception, it is the "atomic set" in that context). The collection of atoms is often assumed to be a set, but this is not necessary, and there are results using the assumption there is a proper class of atoms.
This system is often used to construct simpler independence results concerning the axiom of choice à la what is known as Fraenkel–Mostowski–Specker permutation models.
One of the ways of producing such models is to have the failure of the axiom of foundation (or regularity), in a way where there is a set of different Quine atoms, where Quine atoms are sets satisfying the property $x=\{x\}$. Then we can use the collection of Quine atoms as our urelements, and create a model of $\sf ZFA$. Thus, all methods applicable to $\sf ZFA$ can be applied to $\sf ZF^-$ when replacing the urelements by Quine atoms, and positing that the universe is "somewhat generated" from the Quine atoms (in some canonical way, e.g. $V(A)$ where $V$ is the von Neumann inner model and $A$ is the class of atoms).
This goes in the other direction as well. Starting from a model with urelements, one can create a model of $\sf ZF^{-}$ where there are Quine atoms. So the two approaches are truly interchangeable.
But Quine atoms are not the only way to violate foundation. Indeed, it is possible that the axiom of foundation is violated in all sort of ways, but there are no Quine atoms. And it is possible there is a single Quine atom, as would follow from Aczel's Anti-Foundation Axiom, something which will not be very useful for FMS permutation models.
To sum up, an "urelement" is an object in the set theoretic universe which is not a set. It does not contain any object, as sets contain objects, and urelements are not sets. Sometimes these are called "atoms", which might cause confusion with Quine atoms, but that's not terribly bad, since we can think about them as Quine atoms in a model of $\sf ZF^-$ instead.
So asking if we can construct an urelement which contains Quine atoms is quite a meaningless question. The answer is negative.
Asking if a Quine atom could contain a well-founded set is also not a very meaningful question, since a Quine atom is $x$ such that $x=\{x\}$, the only member of $x$ is $x$ itself, which is not a well-founded set. But it is true that you can have a set which has both a Quine atom and a well-founded set. For example $\mathcal P(x)=\{\{x\},\varnothing\}=\{x,\varnothing\}$ where $x$ is a Quine atom.
