Do the DCT coefficients correspond to particular pixels?
What I'm planning to do is to apply edge detection to obtain the
  locations/indices of the pixels and then get their corresponding DCT
  Coefficient


As others stated in the comment - any pixel in the given block is related to all co-efficient in the block and vice-versa. 

However, understand that Edge detection is a process of finding the gradient. Linear convolution essentially does the job. What you can do is to apply equivalent of that convolution in the DCT domain which will give you an idea about the edge orientation in the DCT block domain.

While, this is relatively narrow area - there are papers which allows you to find edges in DCT blocks. See the references below: 


A Compressed Domain Scheme for Classifying Block Edge Patterns by Hyun Sung Chang, and Kyeongok Kang IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 14, NO. 2, FEBRUARY 2005 145
Direct feature extraction from compressed images by Bo Shen et. al. 
SPIE vol. 2670, Storage &amp; Retrieval for Image and Video Databases IV, 1996