If you strap the mattress to the box spring to stiffen the mattress you can set it on its edge and it won't sag. Then just set the box spring and the mattress on a flat bed bike trailer and you are good to go.

They make trailers like this for big and heavy stuff: https://www.bikesatwork.com/bike-trailers

Okay, I would consider transporting a mattress on a bike (with a trailer) easy, not just a bike!