  We prove that a compact, connected, and oriented 4-dimensional gradient
$m$-quasi-Einstein manifold with $m\in [1, \infty]$ which is additionally a
spin manifold must satisfy the Hitchin-Thorpe Inequality. We show further that
the homeomorphism-type of the universal cover of such a manifold is either
$S^4$ or a connected sum of some number of $S^2\times S^2$ when the potential
function is nontrivial.
