I am building a 14' x 14' gazebo with 6"x6" corner posts. I want to span the distance between the posts without addition support. Can I use a 4" x 6" post or do I need to go to a 2"x10" or 2" x 12" beams? Would I need to double up the 2" beams?

A square hipped roof meeting at one point? Made of what? How much snow load where you live?
(I don't have the answer, but will guess, or you can find roof load calculators online.)

square hip roof - 2 tear. Live in Wisconsin ~50" average snowfall each year. now thinking about just going with 6"x6" beams with anglular supports on each corner post. Think it should be enough structural support to hold up?

6 x 6's 14' feet long? Expensive. What happened to doubled 2 x?
You could gird the posts with 2 x 10s or even 2 x 8s.
And still have the angled supports going inside the 2 x's.
If you don't need a permit, just build it and it's probably plenty of support.
If you do need a permit, submit that and see what the inspector says.

Double 2 X 12 w/ 1/2'' plywood between nail every 6'' both sides. Stagger nails.

Chuck