  We call a metric $m$-quasi-Einstein if $Ric_X^m$ (a modification of the
$m$-Bakry-Emery Ricci tensor in terms of a suitable vector field $X$) is a
constant multiple of the metric tensor. It is a generalization of Einstein
metrics which contains Ricci solitons. In this paper, we focus on
left-invariant vector fields and left-invariant Riemannian metrics on quadratic
Lie groups. First we prove that any left-invariant vector field $X$ such that
the left-invariant Riemannian metric on a quadratic Lie group is
$m$-quasi-Einstein is a Killing field. Then we construct infinitely many
non-trivial $m$-quasi-Einstein metrics on solvable quadratic Lie groups $G(n)$
for $m$ finite.
