A girl finds a dragon egg in a river. At some point in the book she ends up in a mountain trapped with her baby dragons while trying to save a rare dragon egg.
Long shot, but could it be...
DragonSpell (DragonKeeper Chronicles, #1), by Donita K. Paul?
From the Amazon description:
When Kale, a slave girl, finds a dragon egg, she is given the unexpected opportunity to become a servant to Paladin. But on her way to The Hall, where she was to be trained, Kale runs into danger. Rescued by a small band of Paladin’s servants, Kale is turned from her destination.
Feeling afraid and unprepared, Kale embarks on a perilous quest to find the meech dragon egg stolen by the foul Wizard Risto. But their journey is threatened when a key member of the party is captured, leaving the remaining companions to find the Wizard Fenworth, attempt an impossible rescue, and recover the egg--whose true value they have not begun to suspect.
It came up in a Google search for
"dragon egg in a river", and seems to match.