Where did Brian shown confidence in the book Hatchet?

Brian shows confidence in coming up with innovative solutions for his survival. 
For someone who is young and does not have a ton of survival experience, Brian is actually fairly confident.  Many boys in his situation would give up or get discouraged much more easily than he does.  Brian has his ups and downs, but generally speaking he is intelligent and innovative, and continues to work hard.  He doesn’t let mishaps get him down. 
An example of this is his desire to get a better food source.  Brian wants to develop a fish spear.  He makes a plan and carries it out, confident that he will succeed. It is not successful at first. 

He had been so sure, so absolutely certain that it would work the night before. Sitting by the fire he had taken the willow and carefully peeled the bark until he had a straight staff about six feet long and just under an inch thick at the base, the thickest end. (Ch. 12) 

Brian has to “invent” a bow and arrow.  Again, he is confident that this will work.  He doesn’t give up because he is hungry and can’t just go to McDonald’s and order a burger and fries.  When one thing doesn’t work, he knows that he will be able to make something else work.  He believes in himself. 
When Brian misses a search plane, he gets frustrated and feels as if he wants to die.  However, he regains his confidence again.  He realizes that he is clever, and has come this far.  He is confident that he can survive on his own. 

He was new. Of course he had made a lot of mistakes. He smiled now, walking up the lake shore after the wolves were gone, thinking of the early mistakes; the mistakes that came before he realized that he had to find new ways to be what he had become. (Ch. 13) 

Brian fishes out the survival pack, and its contents are almost superfluous because he has been so successful on his own.  However, it has a radio that the search planes use to find Brian.  If Brian had not maintained his calm and confidence, mistakes and all, he never would have been found.