The general and the specific overlap in this instance. Beecher Stowe has provided readers with some pertinent insight into the life of the Quakers and why she chose to incoorporate them into the text. In 1853, one year after Uncle Tom's Cabin was published, she released the Key to Uncle Tom's Cabin. We can spare ourselves historical research and go straight to the text ( electronically available, see link below).
Regarding the Quakers she writes the following:
The writer's sketch of the character of this people has been drawn from personal observation. There are several settlements of these people in Ohio; and the manner of living, the tone of sentiment, and the habits of life, as represented in her book, are not at all exaggerated. These settlements have always been refuges for the oppressed and outlawed slave. The character of Rachel Halliday was a real one, but she has passed away to her reward. Simeon Halliday, calmly risking fine and imprisonment for his love to God and man, has had in this country many counterparts among the sect. The writer had in mind, at the time of writing, the scenes in the trial of John Garret, of Wilmington, Delaware, for the crime of hiring a hack to convey a mother and four children from Newcastle jail to Wilmington, a distance of five miles.
Beecher Stowe, of course, was originally from Cincinnati, Ohio where her father was a preacher. Her memories of these specific incidents are a good example of how the life around her entered her writings.
In general, the Quakers were the first whites to denounce slavery in the United States. Here is an excerpt from historian Christopher Densmore read at a Quaker meeting in New York ( available electronically at SUNY Buffalo Special Collections):
Quakers were forbid any act that acknowledged the right of slavery. Quaker meetings in western New York advised Friends not to use any product-- no cotton cloth, no white sugar-- made with the unpaid labor of slaves. Good Quakers used maple sugar in their coffee. When the government passed laws to make aiding fugitive slaves a crime, the Friends' quarterly meeting in Western New York [Farmington] declared that 'any acknowledgement of obligations to such laws is a violation of our testimony against slavery.'