In First Order Logic if we take the predicate symbol greater, then the formula greater(5,3) is a well-written formula.
If I write greater(x,y), assuming that $x$ and $y$ are variables is it a no well-written formula ?
Or is the presence of a quantifier that defines $x$, $y$ (or any other letter) as variables ?
In first-order logic, you define your language over a signature $\sigma=(P,F,C)$ with a set of predicate symbols $P$, functions symbols $F$ and constant symbols $C$.
Now, for every predicate $p\in P$, you associate with it its arity $\mathrm{ar}(p)$, i.e. the number of arguments it takes.
For any predicate symbol $p$ with arity $n$, and any $n$ terms $t_1,\dots,t_n$, $p(t_1,\dots,t_n)$ is a well formed formula.
Terms may be variables, constants or combinations of other terms using function symbols.
In your case, you suppose that the associated signature contains a predicate symbol $>$ with arity $2$. $$>(5,3)$$ is thus only a well formed formula if $5,3$ are terms in your first order language, i.e. you have $5,3$ present as constant symbols. However, for any variables $x,y$, $$>(x,y)$$ is always a well formed formula as every variable is a term.
EDIT: Don't confuse well-formedness with semantic properties. Well-formedness of a formula asserts that it is part of the associated formal language of first order formulas to a respective signature. This has nothing to do with the property of $>(5,3)$ being valid in some interpretation or not.