In The Fellowship of the Ring, Gandalf says:
It [the ring] has been called that [precious] before, but not by you.
But we know that, at that time, Gandalf didn’t know whether that ring was the One Ring, so how can Gandalf know what he says here?
I mean, he didn’t know that was the One Ring, so he also shouldn’t know that it had been called precious before.
As with most things, this is more clear in the book (emphasis mine):
'It is mine, I tell you. My own. My precious. Yes, my precious.'
The wizard’s face remained grave and attentive, and only a flicker in his deep eyes showed that he was startled and indeed alarmed. 'It has been called that before,' he said, 'but not by you.'
'But I say it now. And why not? Even if Gollum said the same once. It's not his now, but mine. And I shall keep it, I say.'
Fellowship of the Ring Book I Chapter 1: "A Long-Expected Party"
Bilbo had told Gandalf the true story of how he obtained the Ring from Gollum:
I wondered often how Gollum came by a Great Ring, as plainly it was - that at least was clear from the first. Then I heard Bilbo’s strange story of how he had “won” it, and I could not believe it. When I at last got the truth out of him, I saw at once that he had been trying to put his claim to the ring beyond doubt.
Fellowship of the Ring Book I Chapter 2: "The Shadow of the Past"
Presumably, Bilbo also mentioned Gollum's nickname.
It was written in Bilbo's book, There and Back Again, which Gandalf undoubtedly read between the events of The Hobbit and The Fellowship of the Ring. I believe it's also mentioned in Fellowship that Bilbo had originally lied about how he acquired the ring, saying that he won it in the riddle contest with Gollum. (In the original 1937 edition of The Hobbit, Gollum did intend to bet the ring in the game, but Bilbo had already found it. It was changed when Tolkien wrote The Lord of the Rings, to make it more consistant with the Ring's new nature, and the original version was retconned.)
Of course, at the time, Gandalf just figured it to be a simple magic ring, not suspecting it to be the One Ring until much later.
I know this was asked a long time ago but just in case someone comes to this later as I have. After reading the previous posts, I believe that Gandolf comes to the realisation after Bilbo disappearance at his birthday party reminds him of the ring and he connects the disappearance of Bilbo to Bilbos greatly extended life and this connection starts him wondering if there is more to the ring than he believed and that it could posibly be the one ring or at least a dangerous ring, so he decides to then research it?
