No : K3 surfaces admit zero scalar curvature metrics (in fact even Ricci flat ones) but no metric with positive scalar curvature. See for instance the lecture notes by Gromov ([Four lectures on scalar curvature][1]) p. 18.

This comes from the Weitzenbock formula for the Dirac operator and the index theorem.

  [1]: https://www.ihes.fr/~gromov/wp-content/uploads/2019/04/scalar-lectures-IHES-2019.pdf