Rinse quinoa according to package directions (if called for, not all quinoa requires this step). Drain. Heat 1 and 1/4 cup water to a boil in a medium saucepan. (Most packages say to use 1 and 1/4 cup water to 1 cup of quinoa. But go by what your package says.) Once boiling add quinoa, green onions, and broccoli and cook 12 minutes (or according to package instructions for the quinoa)
In a medium saucepan, melt butter. Add flour and stir for 2 minutes. Add milk and bring to a boil while whisking. Once the mixture reaches a low boil, remove from heat and stir in cheeses. Stir until cheese melts.
Drain quinoa and vegetables from the water and add to the cheese sauce. Stir. Pour into a casserole dish and bake 20 minutes. Serve warm.