- Drop the bottom face of the octagon down to the same z location as the plane.
- Then delete the faces on the plane and bottom of the octagon.
You can then select the edges of the bottom of the octagon, as well as the edges of the plane, and F to create a face (which would be an n-gon)...
- Then with the face you just created selected, press Ctrl-Fto bring up the face menu and select "triangulate faces"...
This will result in triangle faces connecting the outer edge of the octagon to the inner edge of the plane.
If you are not happy with triangular faces, you can select the faces in question and use Ctrl-F > Tris to quads
Edit: you may have to join the objects first by selecting both, pressing spacebar, and searching for "join"
EDIT 2: Upon testing my explanation, it didn't work for me as intended... i will leave it there for posterity but ill post the way i did it that DID work.
1: extend the bottom face of the octagon down JUST BELOW the face of the secondary mesh
2: ensure that only the octagon is selected and add a boolean modifier to it [Properties Window > Modifiers > Boolean]
3: Set the "Operation" to "Union" and the "Object" to the mesh in question.
4: Apply the modifier
- note that the boolean modifier will create a copy of the "Object" you selected and work with that, so you may need to delete the unmodified mesh that is left behind from the modifier...
good luck, and sorry if my first answer didn't work =(