There are no overt racism quotes in chapter one of To Kill A Mockingbird. The first chapter does not go into the trial of Tom Robinson or the general racism of Maycomb. What it does instead is introduce the Finches, where they are from and what they are like. It also introduces the children: Dill and Jem. More specifically, the first chapter introduces the mystery of Boo Radley.
If you really wanted to find quotes on racism, you will have to stretch things. Here are a few possibilities.
First, Scout mentions that the people of Maycomb are Southerners and they want to trace their lineage back to England (the Battle of Hastings, no less). This suggests that lineage matters. There is an elitism that might hurt people.
Being Southerners, it was a source of shame to some members of the family that we had no recorded ancestors on either side of the Battle of Hastings.
Second, when Calpurnia is introduced, she is introduced as the housekeeper. However, no mention of her color is mentioned. But later it is clear that she is black and the servant of the Finches. This can be seen as a stereotype. Racism? Perhaps. You will have to wait until the novel develops to see racism.
As was mentioned in the previous post, there are not many overtly racist quotes in Chapter 1. However, Scout does mention racist institutions such as slavery and alludes to racist ideology when describing her hometown of Maycomb, Alabama. When Scout explains her family's history, she describes how her ancestor, Simon Finch, immigrated to America and bought several slaves. Scout says,
"Simon, having forgotten his teacher's dictum on the possession of human chattels, bought three slaves and with their aid established a homestead on the banks of the Alabama River some forty miles above Saint Stephens" (Lee 5).
Clearly, owning African-Americans as slaves involves racist ideology and is an overtly racist institution. Another quote that depicts racist ideology and prejudice towards African-Americans takes place when Scout is describing Boo Radley's history. After mentioning that Boo stabbed his father in the leg with a pair of scissors, Scout describes his punishment by saying,
"The sheriff hadn't the heart to put him in jail alongside Negroes, so Boo was locked in the courthouse basement" (Lee 9).
The fact that white criminals needed to be separated from the black prisoners, as well as the sheriff's apparent disdain for African-Americans, reveals the racism throughout the town.