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Bilbo immediately becomes possessive of the ring, as obsession is a part of the ring's mantra. If you remember, Gollum is just as secretive and protective. Bilbo would keep the ring a secret because he doesn't want it taken away...... because he feels its power and wishes to harness it for himself. These feelings aren't even voluntary..... it's almost as if the ring casts a spell on its owner.
The quote below explains that at first, Bilbo did not mention the ring because he was enjoying praise for his accomplishments.
"Bilbo was so pleased with their praise that he just chuckled inside and said nothing whatever about the ring; and when they asked him how he did it, he said: "O, just crept along, you know-very carefully and quietly."
"Then they wanted to know all about his adventures after they had lost him, and he sat down and told them everything-except about the finding of the ring ("not just now" he thought)."