Both OBB and SBB will be able to sell you a through ticket for the whole route. But you will in each case be travelling on the Swiss system till the border, and the Austrian afterwards. OBB does not operate any trains in Switzerland.
However, the quickest route from St. Moritz to Innsbruck is not via Chur and Buchs, but via Scuol or Zernez to Landeck, and then on the Innsbruck.
In that case if you have a Swiss Travel Pass, or a Saver Day Pass this is going to be valid all the way to Landeck, as the cross border busses between Graubunden and Tirol (Nord and Sud) are included.
You could also go St. Moritz - Zernez - Mals - Meran - Innsbruck, once the Mals - Meran line is operational again.
If you go via Landeck all you would need is a ticket Landeck - Innsbruck in addition if you have the saver day pass.