Unless you have an extremely special 1- or 2-year old they will not know it is their birthday, or that birthdays are times when you have a party and presents.
So don't bother. By all means have a family gathering for you, to celebrate having made this child and kept it alive for a year or two. But if you don't want to then don't. Your child will not know that they are missing out. Do not stress yourself over it. Stressed parents does not = happy babies.