I have a row of Quack grass growing in my flower bed and
I am thinking of getting husband to weedeat it down[no seeds yet] then I will lay paper and mulch over. Good idea or not.
Quack Grass grows by sending out rhizomes as well as be seed so any attempt at control means rremoving every bit of that root material and putting in a barrier to keep it from moving back in. The mulch on your flower bed will make that environment much nicer for the Quack Grass unless a barrier is put in.
I have used Aluminum Flashing (fold the top edge over so no one will slice their hands) inserted vertically into the soil while Janet Macunovich, "Caring for Perennials" uses plastic carpet runners.