For countably infinite graphs you can take the [Rado Graph](https://en.wikipedia.org/wiki/Rado_graph), which contains all countable graphs (even as induced subgraphs).  For higher cardinals, see this [paper](https://link.springer.com/article/10.1007/BF02807219) of Shelah.  As I mention in a comment, the version with minors is not that interesting, since you can just take the complete graph on $\kappa$ vertices.