$L$ is first-order language with identity and $L_Q$ a language obtained by adding to $L$ the quantifier $Q$.
Definition of $Q$: If $P$ is a formula and $x$ a variable, $QxP$ is a formula of $L_Q$. If $A$ is an $L_Q$-structure, then $A\models QxP$ if and only if $|\{a\in A\mid A\models P[a]\}|>\aleph_0$.
Which structure could be model of the formula that is the conjunction of the axioms that say that $R$ is a total order and $Qx(x=x) \land \forall x\lnot Qy(yRx)$
In the real numbers?...but $\mathbb R$??
The real numbers clearly cannot satisfy this formula, because it implies that every initial segment is countable.
On the other hand, consider the ordinal $\omega_1$, it is an uncountable linear order and every initial segment is countable.
Similar models would be replacing every point in $\omega_1$ with any countable linear order, and I believe that you can show that this is really all the models of this theory.