Unfortunately no - the documentation for the API is good and the API really deals with creating, modifying, and closing Test Runs rather than reporting on them.
The reports would be better but you can’t choose the format to see the report.
You could export the Test Run, choose Excel, and then filter the Tested On column. You do have the ability to choose the section or more than one section and what columns you want to see - but no filtering. More labor intensive then you would like I am sure.
Outside of querying the db directly it is the only way I can think of doing what you need. Not sure if that is possible if your install of TestRail is hosted by Gurock.