Isn't feeling here a verb of being so you would use the adjective good?
You could use either, depending on the context.
1) "Feeling well" refers to feeling healthy, not being sick, etc. If someone asks "How are you feeling?", they are generally inquiring about your health, so it's appropriate to answer this way.
2) "Feeling good" is more of a general state of mind, so "I'm feeling good" can refer to someone feeling happy, optimistic, proud, etc. You can say "I'm feeling good about my science project; I think I'll get an A" or "I'm feeling good about helping that old lady this morning." And if someone says "How are you?" and you want to describe a general good mood, you can say "I'm feeling good" (or just "Good").