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Umm, the answer to that question is no.
However, that question is probably not what you actually want to know. Depending on what that is --
No. Merit Badges (custom or regular) are awarded to a particular person, permanently. Given MBs cannot be transferred.
Yes. Per information in Custom Merit Badges and Awardicons from Writing.Com 101 --
The ability to give a custom Merit Badge is not "owned" by a member. The MB belongs to (or is owned by) a Group item. The Group item is owned by a member.
The ability to give a custom Merit Badge is related to being a member of a particular Group item at a particular level of Group membership, which are both set when the MB is created, and cannot be changed. Therefore, a custom MB cannot be transferred to another Group item.
Items can be transferred, so a Group item associated with a custom MB can be transferred to another member, making them the new Group owner who controls which members are able to give that custom MB.
Hope that helps.