This was not true.
It is true that the first abolitionist, society was established in 1777 in Pennsylvania and was a Quaker institution.
Quakers, themselves, owned slaves such as the famous poet Phyllis Wheatley. According to their beliefs as pacifists, their methods were peaceful: sermons, pamphlets, and other means of gentle persuasion. Appropriately, they began with their own meetings.
Quakers lived mostly in the north, but some lived in the future Confederate States of Virginia, North Carolina, and Tennessee, and bore witness in those states.
In 1827, Benjamin Lundy, who was a Quaker, abolitionist, and newspaper publisher, moved to Ohio and began the first abolitionist society west of the Appalachian Mountains. He went on tour and started 130 abolition societies. One hundred were in the South.
Enough said.