Does this wall have any more studs?
Do you know whether the stud spacing in this structure is 16" or 24"? This one stud may be all there is. You may have the edges of studs accessible at the ends of the alcove. If these are available, that would give three studs which is sufficient. Depending on how much weight you plan to put on this, you can use anchors in drywall. ELFA drywall anchors are very secure against pullout stress.

Scan the wall with a stud finder.