No way Connington knows it. Rhaegar was obviously not as close to him as he was to Rhaegar, as it were. Beyond that, let’s see. Howland Reed, Bloodraven, presumably Bran at this point or at some point very soon…I think that’s probably it.
Also, Connington was exiled before Jon was born. He might be aware of R+L, although WOIAF notes that almost no one knew where he had gone, but he couldn’t know about J.