>Was it possible to wage winter warfare in regions with milder climates such as Palestine, Carthage, or Sicily?

The issue is not just weather. You still see cold winters in places like Palestine, but there is also a dearth of food and fodder; many forces throughout history relied on foraging, stealing, buying, or otherwise obtaining food from local sources as they moved. There tends to be rather less of it in winter, which makes supply more difficult. Winter campaigning isn't impossible, but even with a real logistics system, it is more difficult, and there are often other administration issues with maintaining a force year-round. There was an expectation for many forces that, outside of a campaigning season, they would have time to go home, or otherwise find somewhere to become established and start to prepare for the next campaign. Transport, supply, maintenance, etc are all more difficult in winter, and you also have additional costs.

Winter campaigning was done inconsistently for a long time, from the Vikings, to mercenary companies in medieval Italy, to various Roman campaigns, but consistent winter fighting is very modern, and was still influencing plans (especially concerning agricultural workers) in the 20th century. It became less of an issue thanks to industrial and technological factors, so there isn't one point you can definitively identify as the year, but around the 19th century, certainly the latter half, is when armies began to campaign regardless (often with awful results).

This thing is the sort that requires an essay, but in brief:

1. For most of history, war was seasonal: as such, some, if not most soldiers will leave for the winter, and return when the war resumes; those who remain may not have the numbers to launch attacks. Even if the entire army is professional, so skipping planting won't cause starvation, feeding a large force in one place for long is extremely expensive without railroads (and that assumes there's food available at all,) so as much of the army as possible is likely to be dispersed to where they could be fed.

2. Even if the local weather isn't cold, logistics would still be subject to the polar lows, which can lead to sudden and powerful storms: as such, anyone relying on a supply fleet is taking a big risk, and anyone without such a fleet will have trouble sustaining much of a force.

3. The term "Winter Quarters" can also be misleading: while most of the army may not move, small bands of around 50 are very likely to be doing reconnocence and raiding. These raids rarely enter the history books, but they often contribute to depopulating an area, which affects the war.

4. For the most part, winter warfare became more common when steam engines made moving and feeding soldiers much easier. That didn't make it a better idea, though: the logistical issues created by winter still exist, and have still led to disaster when handled poorly.

I live at the foot of the Rocky Mountains, without modern tech it is a giant pain in the ass to get around in the winter. Mud, snow, ice, short days giving way to long nights who's temperatures can kill really makes it difficult for someone to get around, now apply that to a large body of people and supplies drawn by horses. Add in the lack of foraging opportunity and you can begin to understand why they hunkered down for the season.