Could this also be 'They only had a child'?
Sie hatten nur ein Kind und keinen Hund. – They only had a child and no dog [unlike claimed in a previous sentence]. This way, it works, yes. Depends on the context.
Isn't "ein" "a" and "eins" "one"?
Can this mean Sie as in the formal You? i.e "You had one child only". If so, it wasn't accepted.