The following comment is really intriguing!
I would say, "Yes, The equations for going from Mean Anomaly to Eccentric Anomaly to True Anomaly are indeed different for hyperbolic orbits than for elliptical ones, if that's part of your process." The biggest differences are sign-flipping on some of the terms, and the use of hyperbolic trigonometric functions rather than the circular trig functions.
Question: Are hyperbolic trigonometric functions used in calculating hyperbolic orbits? If so, how?