I agree the nitrates are a concern, but the fact is that green/red algae (brush algae is actually red, though it looks more black or very dark green to most of us) is caused by light. It will always find nutrients if light is available.
The aim with a planted tank is to have the light balanced with the available nutrients so the plants out-compete the algae. And the faster-growing plants are better at this because they are using more light and nutrients in order to grow faster.
I would reduce the light period; at the moment it is greater than the plants can use. Every tank is different, so you have to experiment a bit. But reducing by an hour or two hours a day for a week, then another hour if the algae is still increasing, is the way to do it. Don't expect the present algae to disappear, it won't unless you physically remove it. When it no longer is increasing will be the point at which your light duration is basically balanced for the plants.
A "black-out" as we term it can be effective at stopping the algae quickly. But if the balance is not addressed, when the light returns, so will the algae issue.
As to where it came from, there are many sources. Wood, plants, fish from the store. Spores from the air. Algae is everywhere, it was after all one of the earliest life forms that gave rise to all living things, and without algae and plankton we would be in serious trouble.