I don't think there's an established answer to that. Cycling rock (especially with any dead and/or dying organisms on it, whether you can see them or not) may release many things to the water: organics, sources of nitrogen, phosphorus, silicate, trace elements, etc. All of those can help pests such as algae, cyanobacteria, and diatoms.
Mature is in the eye of the beholder. I don't think there's any specific time that always applies, but after 6 months or so, most of these early issue have wound down.
IME, tanks often look best after a couple of years because the intended organisms such as corals and anemones and such have grown to look more natural and fill up the tank, but it hasn't run so long that difficult pest organisms slowly grow to ruin it. The list of those pests is actually fairly long, and many are not easily dealt with.