Yes, proton sticks into a free electron pair on the nitrogen.
Please note that Lewis structures are only an approximation - they don't have to work, especially for compounds containing elements from 3rd period up. At early stages of learning chemistry it is better to accept some things happen (like "polyatomic ions form") and don't get int hows/whys, as you don't know yet enough for a good explanation, and bad explanation (one that is way oversimplified) makes more harm than good.