look into the holes. It looks to me like the bedhead legs have a threaded metal insert that a bolt screws into. Measure it carefully. It is pretty certain to be in millimetres.
the inserts, if you saw them in the factory, might be rivnuts
which are inserted from the opposite end to the bolt, so they don't pull through the hole. If so, you just need a bolt of the same size. It looks like the bolts are inserted and tightened through the big round holes in the ends of the bed sides. In which case you will need hex-headed bolts that you can tighten with a spanner, because I can't see how to get a screwdriver in.
They will be metric bolts.
If the diameter is, say, 8mm. they are called M8 bolts.
The length you want, including that part inside the nut but excluding the head, gives the rest of the description. For example if you need an 8mm bolt 50mm long, it is called a "M8x50 hex head bolt" It does not have a point on the end like a wood screw.
This system is used throughout the entire world apart from three backward nations that do not have much interest in international standards.
The other possibility is that you have Bed nuts. This is a sort of nut with a rounded head so it does not scratch your leg even if it sticks out of the frame. You might not think it looks like a nut, but it is. They are also called sleeve nuts.