It's anybody's best guess really - where the roots of a tree spread to is very much down to local conditions, and the only way you're going to find out is to start digging the hole. At a distance of three metres from the trunk, it's likely you will come across some root material, and losing a little of it shouldn't cause too much damage, particularly at this time of year (assuming you're in the northern hemisphere), but if the whole area is a solid mass of roots, both large, woody ones and fibrous ones, then yes, your tree will suffer and may even die.
Ultimately, if you need to put in a post, then you need to do it, and you may have to replace your tree if putting up a carport is a priority, but you might find it's not an issue when you start to dig out.