Using water from your 10g into your 55g will do very little to help cycle your new tank. It wont hurt to do this, but won't help much either. The beneficial bacteria doesn't live in the water, it lives on the surface of the filter media, sponges, substrate, decorations, the aquarium glass etc.
Are you planning on decommissioning the 10g? If so the best thing you can do is run your HoB
filter in the new tank alongside the canister for 2 or 3 weeks. If thats not possible transfer some filter material or sponges from your old filter into the new. If thats not possible then rinse the sponges from your old filter into the water of your new tank.