Yes.
You don’t say anything about “why” you want to do this and how you think this will improve your stove’s current performance.
Adding this metal alcove and a blower will not make the operation of the stove any more efficient and could actually make it less efficient.
Naturally occuring convection is likely now the best arrangement.
Also, you do not explain what the stove manufacturer or your local mechanical code requires for clearances to combustibles.
and if it can be operated safely in any alcove at all.
While the steel alcove isn’t combustible, it could act as a chimney itself collecting and concentrating heat to the ceiling above which may be combustible.
All in all, what you propose appears to be unsafe and dangerous from here.
You need to consult the manufacturer of the stove and your local code officials before you do anything further.
Your answer may lie in better insulating your garage rather than in trying to reconfigure the stove.
Luck.