Agree with CichlidKeeper01, the beneficial bacteria you are trying to cultivate while cycling are sessile, meaning they live as biofilm on surfaces. They do not live a planktonic existence where you could remove them by doing a water change .
That's true of the autotrophic bacteria colonizing filter media and other oxygenated solid surfaces. However, most bottled cycling products (reportedly) contain the heterotrophic type of bacteria that live in the water column, the same type responsible for 'bacteria blooms'. In other words, yes, water changes don't bother the mature bacteria colony you want to end up with, but they can remove a portion of most types of bottled cycling bacteria, which will be reproducing according to their own ecological niche and conditions in your tank. Add this to the factor that the progress of your true beneficial bacteria colony will vary, and it means the effect of water changes on a new tank can vary with these and other factors.
I've read mixed reviews for Quick Start, don't know which type(s) of bacteria it contains, but when you read their literature carefully it doesn't give you an instant cycle, what it's supposed to do is speed up cycling (according to their literature: complete ammonia removal in 18 vs 31 days and complete nitrite removal in 24 vs 34 days) and keep ammonia and nitrite at reasonably low levels in the meantime. So it appears the theory is to keep your tank relatively safe while your beneficial bacteria colony develops. See
link
In other words, you don't instantly have a mature bacteria colony impervious to water changes. Imo it's best to stock and feed lightly in the meantime and keep an eye on your ammonia/nitrite as you said. Just the eight 2.5 inch fish in a 125 qualifies as light stock and in such circumstances water changes can be normally lighter or less frequent than later on, anyway.
Best way imo to cycle a new tank is with a mature filter if you have enough running filters to spare one, or to temporarily spare one, or seed new filter(s) with some mature media from running filters-- I've done it this way for years, practically foolproof, especially if you stock and feed lightly to moderately at first.