I recently was bought a desk chair because I wanted one that had no arm rests and wasn't too tall. However, it doesn't lean back. Is there any way to make it recline?

Not worth the effort IMO. The amount of time you spend grafting on a reclining mechanism just isn't worth it when you can just go buy another chair with it built in. No clue where you'd even buy the mechanism.

On some chairs, the arm rests are optional. They come with them but they're not an essential structural component.