Pretty much all in the title. When did we first get the suspicion "Hmm, this one might just be the biggest that there is!", and when did we become sure that it truly was the biggest?
"Certain" is not a precise term. As the years have gone by, our knowledge the of earth has grown, and our measuring techniques have become more precise. Thus we have become more and more certain that the peak of Everest is the highest point on the earth's surface. So it is difficult to choose a single point in time.
However, the mountain we now call "Everest" was referred to as "quite probably the highest in the world" shortly after its altitude was first ascertained in 1856.
This measurement was done as part of the Great Trigonometrical Survey, which surveyed the British territories in India. This was a gigantic project, begun in 1802 and not completed until the early 20th century. The area that included Mt. Everest was surveyed in the mid-1800s, under the direction of the GTS's third superintendent, Colonel Andrew Scott Waugh. (Waugh was in the habit of assigning geographical features their local-language names, but this terribly high mountain had no local name that he could determine. So he named it after his precedecessor, the 2nd superintendent of the GTS, Sir George Everest.)
Reference. A report of the above, including Waugh's statement that Everest was "quite probably the highest in the world" can be found in the Journal of the Asiatic Society of Bengal, vol 25 (1856), issue #5. Specifically, see the report of the proceedings of the society, item 7, which is a statement by one Colonel Birch concerning recent progress in surveying. This begins on page 436. [Google Books link] The statement by Waugh, which comes from a letter he wrote, is quoted in the last paragraph on page 438.
P.S. A nitpick. By most definitions, Everest is actually not the tallest mountain the world. That honor usually goes to the island of Hawaii, with its highest point at the summit of Mauna Kea. That isn't a super-duper high peak, but the base of the mountain lies far below sea level, on the ocean floor; it is a very tall mountain indeed. The summit of Everest, OTOH, is the point on the earth's surface that has the greatest elevation above mean sea level. Everest is thus the highest mountain on earth.
The official, recognized height of Mt. Everest was determined by an Indian team in 1955. Since that's the official data being used by Nepal and China, then you can use that year as the time where official certainty was determined.