Yes, flour can be composted.
Flour and the paper bags it usually comes in both compost down pretty quickly so in theory you could just throw them in as they are – but it is better to rip open the bags and mix the flour into the rest of the compost heap for a couple of reasons.
Firstly, rodents love flour and can easily defeat paper packaging – mixing it in will help deter undesirable visitors to your plot. Secondly, a layer/lump of flour is pretty compact/dense and could reduce the air flow in your compost heap – again, stirring it into the heap will lessen that issue.
Also: You can also use out of date flour to make paper glue or play/modelling dough. If it’s infested with weevils, you can freeze it first to kill the little blighters.
So, is flour considered green or brown?
Won’t the flour make the compost smell like vomit? If you leave flour in the sink wet, that smell permeates your house 2 to 3 days later…