It's hard to know where to start with this question because much of the book is about race relations. Since the Civil Rights Movement was a move to try to improve race relations and get blacks to have equal treatment, then everything that has to do with race in the book is connected to the Civil Rights Movement. All of the conditions that black people face in the book are examples of the problems that the Civil Rights Movement was trying to address.
The one place where I can see Malcolm X come in is where the one lady does not want Scout and Jem to come to the black church. Malcolm X was (especially before he went to Mecca) very much of a black nationalist who did not want white help and did not really want blacks and whites to mingle. This point of view is shown in that lady's comments to Calpurnia.