The worlds of Game of Thrones and Vikings have a surprising amount in common, for a fantasy series and a historical drama. Both center on family drama and a battle for the throne (although the 'throne' is more metaphorical in Vikings than the great Iron Throne of Westeros), both feature a lot of fighting, from hand-to-hand combat to massive set-piece battles, and both are elevated by the brilliantly written relationships and personal conflicts.
There are some differences, too, of course - and there are certainly no dragons in Wessex or White Walkers in Kattegat. However, if the characters of one series were dropped in the world of the other, they would probably fit in quite well. In this fictional mash-up, though, who would form friendships... and which would work out, compared to the ones that may start well, but quickly turn (dangerously) sour?
Work: The Seer & Bran
By the end of Game of Thrones, Bran is, essentially, a Seer himself. Of course, by the end of Vikings, the Seer is dead, which would complicate this friendship somewhat! However, the Bran of the later seasons (when he is becoming the Three Eyed Raven) would certainly become a friend and ally to the Seer of Kattegat. The two wouldn't have a traditional friendship, of course, but would likely spend time together in deep meditation, but they would understand each other better than any other pair.
Turn Ugly: Athelstan & Bronn
In many ways, Athelstan and Bronn are similar - they are both flexible, willing to change sides in order to preserve themselves and adapt to new situations. Athelstan may be less cavalier about this than Bronn, and certainly struggles to adapt to the Viking life after Ragnar first kidnaps him from his monastery. However, by the midpoints of their respective arcs, both men are fighters, with a degree of loyalty that can still shift if it is needed to keep them safe. In the end, though, the friendship would fracture, as Athelstan's faith and morality would turn him against the sellsword.
Work: Heahmund & Daenerys
Given Daenerys' arc into a conqueror willing to raze cities to the ground, it would be difficult for her to have any friendship that does not turn ugly - as seen in Game of Thrones, when her closest friends and advisors all turned against her. However, prior to this, she and Bishop Heahmund would probably become allies.
Both are passionate warriors who will do anything for their own cause, and who make a lover of an ally. They would appreciate each other's ability in battle, and devotion to their cause, as well as their willingness to shift original priorities for the person they love. Dany does this when she takes her dragons to rescue Jon beyond the Wall, while Heahmund risks his own safety to protect Lagertha.
Turn Ugly: Floki & Sansa
Floki and Sansa would very possibly struggle from the start, as they are polar opposites for most of their arcs. While Floki is in service to the gods and prefers to be quite isolated, Sansa is happiest in the company of others, and desires to fit in with those around her. Initially, Sansa would likely be quite afraid of Floki, and he would have little interest in pursuing a friendship, but both do later show that they are more than capable of playing the game and manipulating those around them to their own ends. They may be able to use this to work together for a while, but eventually, they would just not have enough in common to work.
Work: Rollo & Jaime
At first, Rollo and Jaime may seem like polar opposites; Rollo is a great warrior, and not too refined, whereas Jaime is a flashy swordsman who uses his name and standing in society to his own advantage. However, these two may become surprisingly good friends over time. Both prove their ability to change, and their desire to stand apart from the expectations of their family name. Both are incredible warriors, and struggle with loving women that they should not and being unable to claim their own children (admittedly, not on anywhere near the same scale). Once past the obvious surface differences, these two would find they have quite a lot in common.
Turn Ugly: Ecbert & Robert
King Ecbert and King Robert would initially connect over their position, ambition, and drive to rule. Ecbert is known for his ambition and vision as a King, while Robert is known for having taken the throne by overthrowing the previous Targaryen King. However, these two rulers would quickly learn that they have less in common than it first seems.
Ecbert is manipulative and cunning, and his ambitions are to expand his kingdom. Robert, meanwhile, has few ambitions by the time Game of Thrones starts. His initial drive was to overthrow a bad ruler, rather than to rule himself, and as a King, he is too trusting and easily manipulated - which Ecbert would not respect.
Work: Bjorn & Jon
Bjorn and Jon would respect each other, especially given their desire to explore, and how they have ruling thrust on them when they do not want it.
Both Bjorn and Jon also have similarities in growing up without a father - although Bjorn did know his father for much of his childhood, he resents him for cheating, while Jon was raised a Snow, and struggles with his place in his family. Both are driven by a desire to do right and a sense of honor, too. They would undoubtedly take off exploring together and not look back.
Turn Ugly: Ivar & Tyrion
Ivar and Tyrion may initially bond over their role as the outcasts of their respective families, and the way that they are underestimated due to their physical differences - not to mention a history of killing their own family members. Both are also incredibly intelligent and cunning, and would likely become allies quickly. However, at their core, Tyrion and Ivar are very different - Tyrion is much more practical, and has a stronger sense of right and wrong. Ivar, on the other hand, gives into his rage and hatred, and has delusions of grandeur that Tyrion would not be happy with. As Tyrion left Dany when her sense of her own power got out of control, he would certainly do the same with Ivar.
Work: Ragnar & Arya
This would certainly be an interesting pairing, but one that would likely work - especially if Ragnar and Arya were to meet at a time when Ragnar was younger, and Arya was trained as the assassin she became toward the end of Game of Thrones. Both of these main characters are incredibly clever survivors, have a truly violent side, are not overly concerned with living up to the world's expectations of how they should behave, and are phenomenal fighters. Ragnar would no doubt be incredibly impressed by Arya's cunning, and she would be impressed by his fighting ability and his own brilliance. Given that Ragnar lost his own daughter early in the series, he may even come to see Arya as a daughter of his own.
Turn Ugly: Lagertha & Cersei
As two powerful women who manage to keep themselves close to power and even hold their own thrones, Lagertha and Cersei may find common ground to begin with. Both can be ruthless, while still being powerfully driven by a love for their family (and especially, their children). However, Lagertha would quickly lose patience with Cersei, and their friendship would end. Lagertha is a very direct woman, and one who does not spend too much time on scheming or manipulating (although she's not above it!) - she is forceful, loyal, and a woman of principle. While she loves Bjorn, she also raises him to be a capable and independent man, and she would be disgusted to see the way that Cersei coddles her own children, and seems to scheme for the love of it. And, as Lagertha is a shieldmaiden and Cersei is by no means a fighter, if it came to a fight between the two of them, there is little question as to who would win.