The fastest legitimate way is to work a small set of honest levers consistently, with no shortcut. Publish genuinely useful content on a focused topic, earn a few real and relevant links and mentions, keep the site technically clean and fully indexable, and stay at it over time. That is the whole list, and the word “legitimate” is doing real work in it: bought links, private blog networks, and aged-domain tricks are excluded, because they are exactly the moves that earn distrust rather than trust.
Useful content on a focused topic is the foundation because it gives Google a reason to index you at all. A new domain has no history, so every page is judged on whether it deserves a place in the index on its own merits. Depth and coherence around one subject let the site read as a credible source faster than scattered pages do. Real links and mentions then act as outside corroboration: when relevant sites and people reference you, that is independent evidence the domain is worth surfacing, and a handful of genuine ones outweigh a pile of manufactured ones.
The technical and consistency levers are the quiet half people skip. A site that is crawlable, fast enough, free of indexing blockers, and cleanly structured removes every excuse for Google to deprioritize it. Consistency, publishing and maintaining over weeks and months rather than in one burst, is what lets trust accumulate, because trust on a new domain is a function of sustained behavior, not a single launch. There is no day count that unlocks it; treat the timeline as a range to watch rather than a deadline to hit.
For your new domain, pick the focused topic, ship genuinely useful pages, keep the site technically clean, and pursue a few real links rather than any hack that promises to skip the wait. The honest path is also the fast one here, because the shortcuts that look faster are the ones most likely to cost you the trust you are trying to build. Work the levers and let the time do its part.