You certainly can install a wood burner in your garden room. In fact, a wood burner is a great way to create a cosy space inside your garden room. Because of the size of garden rooms, you will be limited in terms of what size burner you can install. Smaller burners tend to be more efficient which will save you money. Local regulations apply that determine the type of burner that can be used in some areas.
In most cases a log Burner will also request a 2 mtr external Flue and a hearth to protect the flooring system.