No. The boards are made up of two things.
The configuration of the board that tells it what issues to select, the column setup, colours, permissions and so-on. This is not exportable, although there are a couple of add-ons that are heading towards making the config transferable
The content. The issues in your Cloud instance are not on your Server. So even if you could export the board config, it would not be able to select the issues.
You will need to do some form of synchronisation if you want a board on one Jira to be used in another.
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