Yes, dogs can start having puppies the first time they come on heat, and many dogs do have puppies when they are under one year old as well.
It is of course, preferable, that the female dog be over one year old, so that she can properly care for her puppies, and because by this time she, herself, will be fully grown as well. Having puppies drains the female dogs energy, and dogs that are older than one year deal with this better.
Of course, giving birth and having puppies will reduce your female dog's life span, so unless you are serious breeder, you should not breed her regularly.
It is of course, preferable, that the female dog be over one year old, so that she can properly care for her puppies, and because by this time she, herself, will be fully grown as well. Having puppies drains the female dogs energy, and dogs that are older than one year deal with this better.
Of course, giving birth and having puppies will reduce your female dog's life span, so unless you are serious breeder, you should not breed her regularly.