Yes, but not to destroy it. The only people we know of who were looking for the Ring are Saruman and Sauron.
Most people believed the Ring had already been destroyed in the War of the Last Alliance.
‘I was the herald of Gil-galad and marched with his host. I was at the Battle of Dagorlad before the Black Gate of Mordor, where we had the mastery: for the Spear of Gil-galad and the Sword of Elendil, Aiglos and Narsil, none could withstand. I beheld the last combat on the slopes of Orodruin, where Gil-galad died, and Elendil fell, and Narsil broke beneath him; but Sauron himself was overthrown, and Isildur cut the Ring from his hand with the hilt-shard of his father's sword, and took it for his own.’
At this the stranger, Boromir, broke in. ‘So that is what became of the Ring! ’ he cried. ‘If ever such a tale was told in the South, it has long been forgotten. I have heard of the Great Ring of him that we do not name; but we believed that it perished from the world in the ruin of his first realm. Isildur took it! That is tidings indeed.’
(Council of Elrond)
Only three men escaped from Isildur's ruin to bring tidings of the One Ring:
‘Only to the North did these tidings come, and only to a few. Small wonder it is that you have not heard them, Boromir. From the ruin of the Gladden Fields, where Isildur perished, three men only came ever back over the mountains after long wandering.
Likely this knowledge never spread beyond the White Council, given the Elven tendency to keep matters about the Rings secret. Indeed not everyone knew about the One Ring in the first place - recall that the Numenoreans evidently knew nothing of it.
The Wise who knew about the Ring and knew it was not destroyed believed it was lost forever, having been washed from the Anduin into the Great Ocean. They believed that because, not only was it logical, but because Saruman the Wise repeatedly told them so. Of course, as it turns out, Saruman was himself looking for the Ring. So he may have originally believed this too, but he didn't tell anyone once he suspected otherwise.
’ “At the worst,” said he. “our Enemy knows that we have it not, and that it still is lost. But what was lost may yet be found, he thinks. Fear not! His hope will cheat him. Have I not earnestly studied this matter? Into Anduin the Great it fell; and long ago, while Sauron slept. it was rolled down the River to the Sea. There let it lie until the End.” ’
According to the Appendix B of LotR, Saruman did not begin searching for the Ring until 2851, shortly after dissuading the White Council from attacking Dol Guldur, because Saruman hoped if Sauron was let alone, the Ring would reveal itself. Incidentally, this is 10 years before Thorin and Gandalf set Bilbo off on his Quest. So Sauron and Saruman were actively searching for the Ring at the time Bilbo accidentally discovered it, and as far as we know, they're the only ones who ever looked for it.
We might speculate that Elrond sent some Elves to do at least a cursory check shortly after the Battle of Gladden Fields, but if so, it is not recorded. It would have been dangerous to do so at the time, with a number of Orcs in the area.