$u,v$ are harmonic conjugate with each other in some domain , then we need to show
$u,v$ must be constant.
as $v$ is harmonic conjugate of $u$ so $f=u+iv$ is analytic.
as $u$ is harmonic conjugate of $v$ so $g=v+iu$ is analytic.
$f-ig=2u$ and $f+ig=2iv$ are analytic, but from here how to conclude that $u,v$ are constant? well I know they are real valued function, so by open mapping theorem they are constant?