What is best method to secure heavy curtain rod into drywall?

If the brackets cannot be screwed into a wall stud, take one of the screws from the bracket into the hardware store and look for wall plugs to fit it (there are different sizes available) the plug gets tapped into the drywall and the screw goes into it.

if they're that heavy I'd mount into a stud, otherwise,you can buy anchors that will show how much weight they can carry, consider weight of rod as well

Under the drywall, your windows should be framed in with wooden studs, both a header at the top and "king studs" (isn't that a great name?) on the sides. Try to screw the rod brackets into those wooden studs instead of into the drywall. You should be able to judge the location of the stud by knocking on the drywall, based on where the knock sounds dull and solid rather than hollow, though using a stud finder from the hardware store is the trustiest method.