It's never explicitly stated that Gimli and Frodo discussed the adventures, but,
Glóin and Frodo certainly did
Before the Council of Elrond, after Frodo's recovery, Frodo and Glóin spend a lengthy amount of time discussing the adventures, as well as the future of the members of that party, amongst other things. While discussion of the adventure is never explicitly given, we can infer from the lengthy conversation that other matters had been discussed.
You have had some very strange adventures, I hear,’ said Glóin. ‘I wonder greatly what brings four hobbits on so long a journey. Nothing like it has happened since Bilbo came with us. But perhaps I should not inquire too closely, since Elrond and Gandalf do not seem disposed to talk of this?’
‘I think we will not speak of it, at least not yet,’ said Frodo politely. He guessed that even in Elrond’s house the matter of the Ring was not one for casual talk; and in any case he wished to forget his troubles for a time. ‘But I am equally curious,’ he added, ‘to learn what brings so important a dwarf so far from the Lonely Mountain.’
Glóin looked at him. ‘If you have not heard, I think we will not speak yet of that either. Master Elrond will summon us all ere long, I believe, and then we shall all hear many things. But there is much else that may be told.’
Throughout the rest of the meal they talked together, but Frodo listened more than he spoke; for the news of the Shire, apart from the Ring, seemed small and far-away and unimportant, while Glóin had much to tell of events in the northern regions of Wilderland. Frodo learned that Grimbeorn the Old, son of Beorn, was now the lord of many sturdy men, and to their land between the Mountains and Mirkwood neither orc nor wolf dared to go.
[...]
‘And what of your own people?’ asked Frodo.
‘There is much to tell, good and bad,’ said Glóin; ‘yet it is mostly good: we have so far been fortunate, though we do not escape the shadow of these times. If you really wish to hear of us, I will tell you tidings gladly. But stop me when you are weary! Dwarves’ tongues run on when speaking of their handiwork, they say.’
And with that Glóin embarked on a long account of the doings of the Dwarf-kingdom. He was delighted to have found so polite a listener; for Frodo showed no sign of weariness and made no attempt to change the subject, though actually he soon got rather lost among the strange names of people and places that he had never heard of before. He was interested, however, to hear that Dáin was still King under the Mountain, and was now old (having passed his two hundred and fiftieth year), venerable, and fabulously rich. Of the ten companions who had survived the Battle of Five Armies seven were still with him: Dwalin, Glóin, Dori, Nori, Bifur, Bofur, and Bombur. Bombur was now so fat that he could not move himself from his couch to his chair at table, and it took six young dwarves to lift him.
Book 2, Chapter I: Many Meetings
In later chapters, any mentions of Glóin are merely to describe Gimli as his son, as opposed to discussions between Gimli and Frodo about Glóin and Bilbo. There was, however, plenty of time at Rivendell for such conversations to have taken place, and it is very possible they had.
