Not off-the-shelf. JIRA Service Desk assumes that most requests won't end up with development, so the closest you get is the option for an Agent to click "create linked issue in development project" (The Request and the dev issue then move independently of each other with the agent expected to update the customer as necessary)
You can automate it with some code - I'd reach for automation, script-runner etc to help you with that.
But, I would NOT "move" the issue. I would create a linked one. If you move the issue out of the service desk project, the customer loses all visibility of it - to them, it looks like you've deleted their Request. Never do that to a customer, it makes them angry.
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