"What the parents are, that, to a great extent, the children will be. The physical conditions of the parents, their dispositions and appetites, their mental and moral tendencies, are, to a greater or less degree, reproduced in their children. MH 371"
"Train up a child in the way he should go: and when he is old, he will not depart from it."
The EGW passage directly expands on the principle found in Proverbs 22:6, emphasizing the impact of parental characteristics on children's development. It's a clear application and amplification of the proverb.
"The nobler the aims, the higher the mental and spiritual endowments, and the better developed the physical powers of the parents, the better will be the life equipment they give their children. In cultivating that which is best in themselves, parents are exerting an influence to mold society and to uplift future generations. MH 371.2"
"A good man out of the good treasure of his heart bringeth forth that which is good; and an evil man out of the evil treasure of his heart bringeth forth that which is evil: for of the abundance of the heart his mouth speaketh."
Both passages discuss the influence of one's inner character (or the character of parents) on outward actions and future generations. The EGW quote applies this principle specifically to parents and their children, while Luke 6:45 is a more general statement about the connection between inner character and outward expression.