I've used a "toe ductor" for similar applications https://toekickductor.com/

You're going to need a stack duct (the rectangular duct piece that will extend from the existing floor vent up to the new vent) the same size as the existing vent (i.e. looks like yours is 4" x 12") and a new register collar (which is the piece that the new vent cover will sit on and will attach to the stack duct). Ducts are not usually extended in the way you need for the new nook so attaching/sealing the stack duct to the existing floor collar is going to take a little customization and ingenuity.

Visit your local home improvement store. Find the ductwork section and look at what's available. Try to figure this out.

Thanks all. This is very helpful guidance