Yes. Any complete lattice arises from a Galois connection. For instance, if $(X,\leq)$ is a poset, then the complete lattice associated with the Galois connection induced by the relation $\leq$ is the Dedekind-MacNeille completion of $X$. In particular, if $X$ is a complete lattice, then $X$ is the complete latticed obtained from the Galois connection that we get from the relation $\leq$.

In fact, there are, in general, many ways to obtain a complete lattice from a binary relation, and the ways to obtain a certain complete lattice from binary relations can be completely described.

Suppose $A$ is a subset of a complete lattice $L$. Then $A$ is said to be join-dense if $L=\{\bigvee R|R\subseteq A\}$, and $A$ is said to be meet-dense if $L=\{\bigwedge R|R\subseteq A\}$.

$\textbf{Theorem}$ Suppose that $L$ is a complete lattice and $A,B$
are sets. Furthermore, suppose that $f:A\rightarrow L,g:B\rightarrow
> L$ are functions such that $f[A]$ is join-dense and $g[B]$ is
meet-dense. Let $R$ be the binary relation on $A\times B$ such that
$aRb$ if and only if $f(a)\leq g(b)$. Then $L$ is isomorphic to the
complete lattice obtained from the Galois connection induced by $R$.

The above result is the best possible in the sense that the only binary relations $R\subseteq A\times B$ that give you the complete lattice $L$ are induced by functions $f:A\rightarrow L,g:B\rightarrow L$ with $f[A]$ join-dense and $g[B]$ meet-dense. In fact, we obtain a duality between the class of all binary relations and the class of all triples $(L,f,g)$ where $L$ is a complete lattice, $f:A\rightarrow L,g:B\rightarrow L$, $f[A]$ is join dense, and $g[B]$ is meet-dense.

One should probably look at any reference on formal concept analysis for more information on the correspondence between binary relations and complete lattices. The book Introduction to Lattices and Order by Davey and Priestley has a chapter on formal concept analysis.