Can one migrate own question from math.stackexchange.com to mathoverflow.net or Conversely? Could this be a help for moderators?
If I understand your question correctly, you are asking whether the OP can migrate question by themselves or whether some intervention by the moderators is needed.
The question cannot be migrated by the OP.
If you want to migrate your post to MO, you should flag the question for moderators attention and explain in the flag that you want to migrate the question. (Maybe you could also explain the reasons why you want to migrate the question.)
Don't forget that the questions older than 60 days cannot be migrated.
(A question can also be migrated to another site without moderator intervention, if enough users vote to close it as off-topic and choose to option to migrate. But MathOverflow is not among the migration paths on this site.)
For more details about how migration works see What is migration and how does it work?
If you decide to repost your question on MO rather than migrate, it is recommended, that you clearly state in both posts that the same question is asked on the other site and link to the other post, see here. Both cross-posting and migration have some advantages and disadvantages, see this discussion: Cross posting, or flag for migration?