None.

Whatsoever.

Anyone who tells you otherwise is either grossly misreading the situation or is trying to push some other agenda.

This sort of theory falls down at the first hurdle even. As you note, the "Viking Age" starts with an attack on England, and picks up with large scale raids, mostly on England, who, very obviously did not attack the Saxons. Nor were the responses timely, the attack on Lindisfarne in 792 came nearly a generation after the conquest of Saxony by the Franks under Charlemagne. While the Saxon Wars did continue on for several decades, it would be a questionable strategy for Norse polities to aid the Saxons in their conflict against the Frankish armies by attacking English monks in Northumbria. Isolated attacks in this vein continued for much of the 8th century and the early 9th century, small and sporadic attacks on isolated and weakly defended centers of wealth (both loot and slaves) in the British isles and around the Baltic.

Viking attacks on the Franks, who were to be clear the ones attacking the Saxons, not the English, the Irish, the Picts, or any other group of people, only tick up after the reigns of Charlemagne and his son Louis the Pious who both led powerful centralized and effective governments that were able to keep the Frankish heartlands secure from Norse raids (for the most part).

The motive of Norse raids were absolutely economic in nature, they wanted wealth, in cash, kind, ransoms, or slaves.

The Viking depredations on the rest of Europe were not caused by any one single factor admittedly, but certainly not due to Christianization of a neighboring people (that they occasionally fought with as well). Rather they were the result of economic opportunities that the Norsemen were able to take advantage of. This did not always take the form of raiding though. Trade, mercenary service, "colonizing", and raiding were all essentially the same goal, economic advancement, just by different means. In many ways the Viking era was just an extension of the earlier migration period, so long as you buy into the existence and importance of relatively large scale migration in Europe from the Germanic World (and later the Slavic world). The fundamental reason behind migration is usually economic opportunity. In the late Roman world this took the form of Foederati service in the Empire, raiding into the Empire, exacting tribute from the Romans, and so on.

The viking playbook was quite similar in many ways. Norsemen exacted tribute from those they raided, a famous example is in the Viking attacks on Paris and Francia. They attacked valuable targets such as monasteries for their collections of valuable artifacts (and people). However this was not the entirety of their actions with the rest of Europe. They were hired out as mercenaries, most famously in the Varangian Guard of the Byzantine Empire. Trade opportunities were seized, particularly in Russia and Finland, as a part of the lucrative trade with not only the Byzantine Empire but with the Muslim world as well. And as you note, colonizing efforts were launched in some areas, particularly Ireland, the Danelaw in England, Novgorod and Kiev in Russia, and Normandy in France. However there is no general consensus on the actual numbers of Scandinavian migrants who arrived in these locations. Nor is there consensus on the population make up of these areas, ie did women and children come to live in these areas or was this a male only phenomenon.

I'll unpack each of these a little.

Raiding is a pretty straightforward action. Show up in an area, take what you can in movable wealth and bring it back home. This could take the form of gold, silver, and other precious materials, slaves, and so on. These raids happened all over Western Europe, and some into the Mediterranean. The "Great Heathen Army" that overran England is often seen as the climax of this phase of the Viking Age. The Great Heathen Army was perfectly fine to take tribute and gifts from native Anglo-Saxon kingdoms, storied of Ragnarr Lothbrok and oaths of vengeance notwithstanding.

However this period of raiding and returning home eventually gave way to more organized efforts at conquest, such as adding England and other areas to the Norse possessions back in Scandinavia. Peter Heather argues that this was due to the increased involvement of powerful rulers in Scandinavia as opposed to just small warbands that were incapable of taking on larger realms or polities. However even in this period Scandinavia was quite a motley assortment of small polities, unification in Scandinavia would only come about towards the end of the Viking Era. This increased consolidation coincided with larger and more ambitious adventures at attempts at outright conquest. Danish conquests in England under Sweyn Forkbeard and Cnut the Great come immediately to mind in the early 11th century. This was after the period of the "great heathen army" and after the kingdom of Wessex had unified England. Harald Hardraade's invasion of England in 1066 also comes to mind.

Mercenary service is a little less visible in the historical record. It has a clear historical precedent however in the foederati of the late Roman Empire and the service of Angles, Saxons, and Jutes in Britain before the Anglo-Saxon conquest. The Varangian Guard of Byzantium is the most famous example and easily the most well attested mercenary group at this time. Membership in the guard was quite a prestigious, and profitable, appointment. Sagas mention the splendorous wealth of those who served in the guard and returned to Scandinavia. The Varangians supposedly had the right to carry all they could from the royal treasury upon the death of the Emperor, this is mentioned in Harald Hardraade's saga, and that would be in addition to their normal pay and plunder they would win in their involvement in Byzantine affairs.

Trade was also a part of this system. Goods from the Islamic world have cropped up in Scandinavia such as silver coins and cloth, likely through the intermediaries in Russia (also Norse dominated at this time). Peter Heather argues that Norse exploitation in Finland and Russia was another extremely profitable endeavor for Norsemen, especially in the trade of lumber and furs with the Islamic World. This Scandinavian domination of Eastern Europe eventually gave rise to the Russian states of Novgorod and Kiev.

Now there's "colonization/migration/whatever you want to call it and this is inextricably linked with trade but I am not super familiar with the specifics of this aspect of Norse life. Norse rulers ended up in positions of power and influence in many parts of Europe that previously had little Scandinavian influence, but to what degree the Norse imposed themselves on existing frameworks and how much as their own innovation is up for debate. Certainly many of the areas that they settled were already long inhabited and involved in trade networks. Calling it a process of colonization is a little controversial, and presumes that there was an outside imposition on indigenous ways of life. This cannot be confirmed definitively and this process could have been a more migratory process with Norsemen taking advantage of economic opportunities elsewhere and locating themselves there.

In some cases the Norse simply came to dominate local politics and trade, ie Russia, Normandy, the Danelaw which indicates more of a migratory pattern than colonial. Other places they settled wholly, Iceland, the Faroes, and then there's the famous Vinland Saga. I cannot really speak to Norse involvement in Ireland or Scotland so I'll leave that for someone else to talk about. Trade in these areas was clearly quite profitable and far reaching. Silver from the Islamic world has found its way to Scandinavia, likely in exchange for lumber and furs.

Viking raids were merely one aspect of viking economic opportunism across Europe in the middle ages. This was not a case of vikings deciding they didn't like the Saxons being attacked by the Franks so much as it was a process by which Norsemen were able to seize on opportunities that existed across Europe. This took the form not just of raiding but also mercenary service, trade, and migration/colonization. In some ways this could be seen as just a continuity of the earlier Germanic migrations that marked Late Antiquity.

However your question gets to a broader point. There was no assumed cultural kinship between the Saxons and the Norse. They were not co-coreligionists fighting against interlopers, indeed their shared cultural elements were hardly a uniting force. The social world of the Early Medieval World was not rooted in shared identity, be it culture, language, or religion. Identity was tied at this point to personal allegiance and reciprocal social systems, not shared culture. The Norse and vikings would not have leapt to the defense of the Saxons because they had no reason to do so. They owed them no allegiance nor did they have any reason to see themselves as belonging to the same group of people.