How to cite a specific organization in APA?
In the text:


  "This aggression will not stand," confirmed Bob Loblaw, attorney at the Texas non-profit organization Trial Lawyers for Public Justice (Loblaw 2018).


In the references section, you will write up whatever medium you found the quotation in, and TLPJ might or might not be explicitly mentioned.


If it is a blog post or technical report or such authored by Loblaw, hosted on the TLPJ site, you would have Loblaw as author and mention the organization only if it is in the overall blog name or as the distributor/publisher of the report.
If there's an unattributed press release from the organization, within which is Loblaw's statement, Trial Lawyers for Public Justice is the author to list. 
But if Loblaw was quoted in a newspaper, say, then you cite the newspaper article in the references, and your text would be something like:


  "This aggression will not stand," confirmed Bob Loblaw, attorney at the Texas non-profit organization Trial Lawyers for Public Justice (as quoted in Lane 2018).