Absolutely NOT.
The groundING conductor (green or bare) is bonded to the ground bus in the panel - regardless of whether the circuit is protected by an "ordinary" overcurrent protection breaker, a GFCI breaker, or an AFCI breaker.
A GFCI breaker (or an AFCI breaker) will have a neutral conductor from it that must be attached to the neutral bus in the panel.
Both hots from the circuit's cable attach to the hot terminals on the breaker, and since there is no neutral conductor for the circuit, there is nothing to attach to the neutral terminal on the breaker.
It is only the hots and the neutral (groundED) conductors that must pass through the GFCI breaker. The ground (groundING) conductor has nothing different to its installation than any other situation.
GFCI protection works on detecting any current that does not return through its intended path. With a single pole breaker, this means measuring the current through the hot and the neutral to detect a difference. With a two pole breaker, current in one hot must return through either the other hot, or the neutral, or split between the two. GFCI protection involves measuring all three to make sure everything works out even.