There are three "Caucasian Nations", and two of them were fighting. But there is almost no news about the other one - Georgia. What did it do during the war? Did it support any of the two, whether publicly or secretly? Which of the two get more support from normal Georgian people?
It's not clear that Georgia[ns] did anything in particular other than worry. And that's because Georgia has sizable minorities of both Azeris and Armenians, so they worried the war might drag these minorities into ethnic strife on Georgia's territory:
Georgia’s ethnic Armenian and Azeri minority communities are the largest in the country at 168,000 and 284,000 respectively. Although there are cases of coexistence and even cohabitation in several villages, towns, and cities, most communities remain largely separate, which is why some analysts are starting to express concern.
Demonstrations have already been held, with ethnic Azeri citizens of Georgia marching on the Azerbaijani Embassy in Tbilisi to show their support for the military action. [...]
Other than that, Georgia was unhappy with the West not having much role, so in a sense giving Russia more clout:
“Of all the non-belligerent regional actors, Georgia has the most to potentially lose from this war”, says Eurasia Democratic Security Network fellow Alexander Scrivener. “It is clearly unrealistic for Georgia to take on a leading role in mediation and it should probably avoid involvement in any physical peacekeeping mission.” [...]
“This war is bad news for Georgia”, he concludes. “But just how bad depends on how the situation develops. At a minimum, Georgia faces the prospect of dealing with the economic effects of being surrounded by a wall of fire. Conflict to its south, in addition to its already fraught situation with Russia, is not good news for a Georgian economy already reeling from Covid-19".
“The conflict, by highlighting the West’s impotence and disinterest in the region, is also likely to lead to further entrenchment of Russian hegemony here”, he says. “This, to put it mildly, is also not good news for Georgia”.
And indeed Georgia was not involved in any way in the armistice etc. That was negotiated and upheld by Russia and Turkey, the patrons of the two parties in the war.
Actually, Georgia did try to broker peace talks at one point, but these efforts were rebuffed:
On September 30, three days after the fighting broke out, Georgian Prime Minister Giorgi Gakharia invited the warring sides to talk it out in Georgia. “Georgia is ready support the [peace] process in any way possible, including hosting a meeting of the parties to the conflict in Tbilisi to hold a dialogue,” he announced. [...]
“That offer was essentially meant for Baku, which, as always, is not willing to negotiate,” Armenia’s ambassador to Tbilisi, Ruben Sadoyan, told the news service Kviris Palitra. [...] Azerbaijan’s envoy, Faig Guliyev, said that Baku would only agree to negotiate if Armenia agrees to pull out its troops from breakaway Nagorno-Karabakh and the surrounding territories.
And Georgia tried to project an image of neutrality by taking some measures against both sides:
To pre-empt accusations that it is taking sides, Georgia moved quickly to close its borders and airspace to military supplies to both Armenia and Azerbaijan. “Both sides were informed of this,” said Georgia's National Security Council in a statement, adding that non-military transit continues as usual.
Generally speaking, it seems Georgia's position has always been that Karabah belongs to Azerbaijan though (in line with the UN view). On top of this, Georgia has more extensive economic relations with Azerbaijan than with Armenia, but also had some localized disputes with Azerbaijan:
Georgia’s ethnic-Armenian community protested the “pro-Azerbaijani” statement of former Georgian president Mikheil Saakashvili, who emphasized that Karabakh is the sovereign territory of Azerbaijan (Civil.ge, 1tv.ge, October 1; Resonance daily, October 7). [...]
Despite Georgia’s friendly ties with both Armenia and Azerbaijan, the relative quality of these relations differs based on several significant factors. Armenia is a member of the Russia-led Eurasian Economic Union (EEU) and Collective Security Treaty Organization (CSTO), while Georgia is a North Atlantic Treaty Organization (NATO) and European Union aspirant country. Its relations with Azerbaijan are more strategic. Economically, Georgia is more extensively linked to Azerbaijan than Armenia. About 95 percent of Georgia’s natural gas comes from Azerbaijan (Factcheck.ge, June 17, 2019). [...] A prolongation of the conflict could also jeopardize the security of important regional oil and gas pipelines, including Baku–Supsa, Baku–Tbilisi–Ceyhan and the South Caucasus Pipeline.
That said, Georgian-Azerbaijani relations are not cloudless. The Christian Orthodox monastery complex David Gareja (Keshikchidag to Azerbaijanis), which is located along both sides of the conditional Georgian-Azerbaijani borderline, has become a subject of a border dispute leading to several localized incidents and clashes (Agenda.ge, October 10, 2019; see EDM, May 14, 2019, June 6, 2019, July 30, 2019). Currently, the issue is being discussed by a bilateral commission, and the parties claim they will be able to resolve it based on consensus (Agenda.ge, September 24). However, Baku’s future attitude to this matter may very well depend on the outcome of the ongoing conflict in Karabakh.