Were they generally supportive or against the Austro-Hungarian government?
So, there is a bit of a problem here in that there isn't really an average Galician. Galicia was a multi-ethnic society which formed into often opposing political blocs. Likewise, the Austro-Hungarian Empire existed for half a century and had been part of the Austrian Empire since the eighteenth century. So was there a specific period or group you were interested in? I can, however, make some broad statements, and given I know the period around the First World War the best I'll focus there.
In 1914, it seems that yes most Galician factions supported the Dual Monarchy, especially when the alternative was conquest by Tsarist Russia.
Galicia, the territory in which the army was to operate, was a centre of seething national ambition and conflict. The struggle for power between the Polish-dominated administration and Ruthinian nationalist intelligentsia was, despite concessions, still fresh and bitter in 1914. The stakes in the political competition had in fact become even higher once war broke out, and both peoples' representatives tried to curry favour with the Emperor and win political leverage by raising armed forces, the Polish Legions and Sič riflemen. Both peoples were mostly loyal to the Habsburg state, in good part because they needed its support in their conflict. Among the Poles, even Ignacy Dazyński and his Socialist comrades had approached the government in Vienna at the start of August and optimistically promised an uprising in Russian-held Congress Poland, claiming to have tens of thousands of revolutionaries prepared and waiting for the moment to attack the Tsarist oppressor.^157 Only the National Democrats, who had their base in eastern Galicia, were tactically pro-Russian, but even they were quiescent in the summer of 1914. The Ruthenes, as the weaker people, were even more reliant that the Poles on Vienna's support. The majority Ukrainophiles — the nationalists — fervently proclaimed their allegiance to the Empire's war effort in 1914. However, this people's public image had been tarnished by the pre-war spy scandals and accusations, unjust for all but a small part of the population, of Russophilia.^158
This seems to be a actually seems to be a general trend through a lot of the Empire's history. Competing ethnic groups might fight among themselves but looked to Vienna for support and hoped to increase their standing in the Empire rather than to separate from it.
However, this system of reliance on Vienna, just those around the rest of the Empire, did not last through the war. The mood seems to have been particularly pro-Habsburg through out 1914 and '15 and the promise of an independent Polish Kingdom kept popular opinion generally good through 1916 too but by 1917 things started to get quite bad. The decline in enthusiasm has many roots, such as the perceived mistreatment of the Polish Legion, There seems to be two main factors in the decline of popular enthusiasm and through poor handling both came to a head at the same time. Economic conditions in Galicia had seriously declined throughout the war and in the spring of 1917 food shipments were diverted from the province to feed Vienna just as central powers plans for post war Eastern Europe became public.
The news that Chelm would go to Ukraine caused outrage in Galician society. The politicians of the Reichsrat's Polish Circle bitterly denounced the treaty. National Democrats and Socialists were especially scathing. Ignacy Dazyński, the leading Socialist light, declared 'the star of the Habsburgs [to have] gone out in the Polish sky'. The conservatives were at first more hesitant about a total break with the Monarchy, but the later revelation of the secret agreement to divide Galicia administratively between Ruthenes and Poles alienated them too. After fifty years of loyalism, Polish politicians had been pushed into opposition by Czernin's catastrophic diplomacy.^24 In sharp contrast to what was perceived as Habsburg betrayal, the Allies had raised their ideological bid for the Poles' support.
By 1918 this discontent had moved beyond opposition and into lawlessness, armed resistance and eventually revolution that would bring down the Empire.
Source:
Watson, Alexander, Ring of Steel: Germany and Austria-Hungary at War, 1914-1918 (UK: Penguin Books, 2015)
Now, Alexander Watson is basically stating the consensus here so I don't feel bad about relying solely on him but I do wish I wasn't currently stuck at home thanks to the COVID situation and could have used a couple of other works from my University's Library. If you are interested in checking them out, however, i strongly recommend Mark Cornwall's The last years of Austria-Hungary: a multi-national experiment in early twentieth-century Europe and István Deák's Beyond Nationalism: A Social and Political History of the Habsburg Officer Corps, 1848-1918. John Schindler's Fall of the Double Eagle: the Battle for Galicia and the demise of Austria-Hungary might also be of interest.