The Thinkpad A31 (other models too, and some Dell Latitudes) use a special housing for the optical drive.
If you have the old optical drive then you can remove the housing and mount it to the new BD writer. That should work fine.
If you don’t have the old optical drive then you have to buy a cheap optical drive specifically for Thinpads and remove the housing from that.
Here is a video of doing this for a Dell, but it should be roughly the same for a Thinkpad: