No. Jag sätter mig means you're sitting down = moving from standing to sitting position and Jag sitter ner (ner is unstressed) means you are already sitting down, no movement involved.
However the command form sitt ner normally means 'sit down' as in the movement. (it's possible for it to mean 'remain seated', and it's also possible to say sätt dig/er ner)
I translated this sentence into English as: "I sit down IN the chair." But this was not accepted. Apparently one must say "ON the chair," not "IN the chair"? To my ears, it is more natural in English (or at least equally so) to say "IN the chair." Of course, there are chairs that one can only sit ON, but they are literally stools. So... could we get "IN the chair" accepted, please?