This question already has an answer here:
Which metrizable topological spaces $(X,\tau)$ posses the following property:
Every compatible metric (i.e one which induces the same topology $\tau$) is complete.
Compact metrizable spaces satisfy this.
Are there any non-compact examples?
It turns out that if you impose additional structure the answer is yes. In particular, if $M$ is a smooth manifold, and every Riemannian metric on $M$ is complete, then $M$ must be compact.
Reference: Nomizu, Katsumi, and Hideki Ozeki. "The existence of complete Riemannian metrics." Proceedings of the American Mathematical Society 12.6 (1961): 889-891.