I have a metal patio cover over cement. Would like to brick around it, 4 ft high, to make it a room but not complete enclosed. What would be the best material to use and the cheapest.

"Brick around it" If that was your plan, why ask what is best and cheapest? Brick is probably best and most expensive, PT frame wall and Hardie is probably next and a vinyl wall screen from roofing material is cheapest-and probably the worst looking.

Do you know if I need a permit? What type of brick would work best? How to figure out how many bricks needed? How work I connect the bricks, mortar?

What would be the best type of brick? Would this require a permit? Can I stack them without any type of mortar? First experience with trying to build something out of brick.

Duplicate comment sorry new to this - How could I, not How work I

I wouldn't suggest "stacking" up to 4ft, any person trips and falls, or even leans, on that and it will go.

to work out the number of bricks you need the easiest way is to work out the surface area of each wall (height x length) and then divide that by the surface area of your brick ((height + 10mm for mortar) x (length + 10mm for mortar)). That way you'll know how many bricks you'll need to make a wall that size.

If you've never laid brick, you may be surprised how difficult it is. You will need a "hard" brick, one designed for outdoor use, decide how you would run the pattern, mortar the bricks, strike the joint, etc. I think you would find it a lot easier to build a PT frame and side it with Hardie, either sheet or siding. Lot easier, cheaper and it would look good if you painted it a matching color. As far as a permit, we have no idea where you live, North or South pole, equatorial Africa or?? You need to check with you local permitting office.