Indaswamp
might be right, but the end with the orifice looks to me like a straight, not a tapered thread. It appears to seal on the burner/orifice end with an O-ring slightly larger than the orifice insert, so the threads needn't be tapered to create a seal.
So I'd assume that end could be any weird thread and proprietary connector, so take good care of it...might be hard to replace. But the
other end of that short brass burner fitting is almost certainly 1/4 NPT female. So you really just need an extension from 1/4 NPT male to female, in the added length you want, to insert between your hose and the somewhat unique brass fitting with the orifice.
Will
5-foot do the trick? Note male-to-male hose is much more common so a
1/4 NPT coupling will be needed if you get one of those.