As
@CrazyWizard has said, hovering over the inactive Tournament button on the world map will tell you what the next tournament is.
On Mobile, however, you can get that information by checking the Tournament rankings from the Rankings window (
not the fellowship window)
Trying this on PC won't tell you that info. It
will tell you what the previous Tournament was (so long as one isn't currently running) so you can infer what the next tournament will be since it's always marble, steel, planks, crystal, scrolls, silk, elixir, magic dust, and gems. In that order. In the order it appears in the Goods drop-down menu on the top bar.
You can also make the same inference if you were in a fellowship during the previous tournament by checking the Tournament rankings from your Fellowship Window, where it'll tell you what you got from the prizes awarded as a group (as opposed to the individual awards gained from beating up provinces). The relics earned from those chests will always match the flavor of tournament it was earned in. In this case, it was steel. So the next will be planks. The following image is from mobile, but the same info can be gotten from the PC version. Just not quite as pretty as mobile.