It won't speed up the process... The bacteria you want is on surfaces rather than in the water column. So the water won't do much unless you have ammonia in it (which hopefully you won't given your 120 is cycled and healthy). However, if you are stirring up detritus that is captured in your water changes then it can be a source of ammonia to help your 300L along without having to add another ammonia source -- but it is far easier to either use pure ammonia or just ghost feed.
What will help speed up the process is if you seed your new tank with media, sand, and/or rocks from your established one. That will give a good amount of bacteria as a start point...then all you need is wait until they bacteria multiple enough to support a 300L system. Hope that makes sense...