As Dottard and agarza pointed out, oak trees were large and provided excellent shade. Here are some more examples from the OT:
And Joshua recorded these things in the Book of the Law of God. Then he took a large stone and set it up there under the oak near the holy place of the LORD.
The angel of the LORD came and sat down under the oak in Ophrah that belonged to Joash the Abiezrite
Gideon went inside, prepared a young goat, and from an ephah[fn] of flour he made bread without yeast. Putting the meat in a basket and its broth in a pot, he brought them out and offered them to him under the oak.
and rode after the man of God. He found him sitting under an oak tree and asked, “Are you the man of God who came from Judah?” “I am,” he replied.
They sacrifice on the mountaintops and burn offerings on the hills, under oak, poplar and terebinth
Now back to Genesis 35:
4 So they gave Jacob all the foreign gods they had and the rings in their ears, and Jacob buried them under the oak at Shechem. 5Then they set out, and the terror of God fell on the towns all around them so that no one pursued them.
6Jacob and all the people with him came to Luz (that is, Bethel) in the land of Canaan. 7There he built an altar, and he called the place El Bethel, because it was there that God revealed himself to him when he was fleeing from his brother.
8Now Deborah, Rebekah’s nurse, died and was buried under the oak outside Bethel. So it was named Allon Bakuth.
An oak tree was big. It served nicely as a sort of address or location identifier. Also, it provided excellent shade to sit under or dig.
Is there a correlation between Genesis 35:4 and Genesis 35:8? If so, what is the spiritual meaning behind it?
'Under the oak's were just practical convenient locations. I doubt there is any spiritual significance.