Rinse quinoa according to package directions (if called for, not all quinoa requires this step). Drain. Heat 1 1/4 c water to boil in a medium saucepan. Once boiling add quinoa, green onions, and broccoli and cook 12 minutes (or according to package instructions)
Meanwhile, chop onions and broccoli and shred cheese. In a medium saucepan, melt butter. Add flour and stir 2 minutes. Add milk and bring to a boil. Once mixture reaching a low boil, remove from heat and stir in cheeses. Stir until cheese melts.
Drain quinoa and vegetables from water and add to cheese sauce. Stir. Pour into a casserole dish and bake 20 minutes. Serve warm.