Yes adding water from an established tank will help. Although the nitrifying bacteria is predominantly sessile, bacteria is everywhere so the old water will have a lot of beneficial bacteria in various stages of reproduction/death present. Adding established substrate and established filter media into your filter will also reduce the biological cycle establishment time. I recommend you clean out a dirty filter and empty the collected effluent into your new tank. Although it makes your new tank dirty for a day, the substrate, ornaments, glass etc AND new filter will be full of bacterium.