This is a translation of a Greek phrase, aphthiton kleos, which refers to the undying glory sought by epic heros. In this context, it refers to the glory which can be won during hunting in the English countryside, instead of the fame which can be achieved during battle. In this case Mark Robarts is said to have won “immortal glory” among his hunting companions for his performance during a hunt. Lady Lufton very much disapproves of this behavior from a clergyman and is thus not pleased when she discovers this information. [MD 2005]
Sources: Homer, Iliad 9.413 (for an example of the use of the phrase).