I personally prefer the phrase "from the Vikings to the Crusaders" to "from the Vikings to the knights", but there was certainly such a general trend in medieval Scandinavia in course of the 12th to the 13th centuries.

While more can always be said, I hope some of the following previous posts of mine might be interesting to you:

• Would post-viking era Scandinavian armies and soldiers be organized and equipped any differently than other "western" medieval armies?
• When did the Shield-wall strategy die?
• At What Point Were Vikings no Longer Considered Vikings, and Just Seen as Christian/Christianized Danes, Swedes, Norwegians, etc.?
• When did tensions between the people inhabiting modern day Britain and Scandinavia subside following the raids/invasions of the 8th-11th century?

+++

First of all, please look at this carved wooden door that depicts the mounted knight (center) fighting the dragon (right) to free the lion (left).

This door, now held in the National Museum of Iceland (and one of my strong recommendation among their collection), is taken from the old church at Valþjófsstaðahurð, Eastern Iceland. Usually dated to about 1200 (prior to the very famous manuscripts of Eddas that record Old North myths!), however, it is often suggested that this door had originally used in the local farmstead nearby the church.

Researchers have debated which episodes (note the plural) of the "sagas" are most well applied to the iconography of this engraving for long (Harris 1970), but they presuppose in common that this kind of the friendship between the knight and the lion indeed became popular in the 13th century Iceland, thank for the reception of the translated chivalry literature like (Ívens saga - Old Norse adaptation of Chrétien de Troyes' Yvain ou le Chevalier au lion (about 1180)), about the same time when Snorri Sturluson (and his scribes) put Eddic stuffs down on the parchment.

12th and 13th centuries are generally regarded as a kind of transitional period also among the aristocrats in Scandinavia:

• From the second decade of the 12th century onward, we don't have almost any positive written evidence that support the use of shield wall strategy in actual battles. Instead, heavy cavalry sometimes began to be mentioned as game-changers in the battles in contemporary Scandinavia.
• About the same time (from 1130s onward), almost the all contemporary history and literature texts in Europe ceased to regard Scandinavia as a homeland of the Vikings, possible threat.
• Some popular chivalry literature works are indeed supposedly translated at the court of King Håkon IV of Norway (r. 1217-63) under his auspice in 1220s (?).
• In later 13th century, aristocrats both in Norway and in Sweden were now called with the title "the knight" (ridder/ riddar) either in documents or in law codes.
• From the 13th century onward, several Scandinavian aristocrats are known to employ the mounted knight as their sigil (this famous sigil (sorry for the low quality reproduction) is that of Jarl Birger Brosa of Sweden (d. 1202)).

So, I'd say with relative easy that the chivalry ideal and courtly culture got foothold among the Scandinavian (and even some Icelandic) elites in the High Middle Ages. They adapted these "European" culture as a part of their status symbol also in Scandinavia.

I'm also willing to answer any additional question on the relevant topic.

Additional References:

• Bandlien, Bjørn. "A New Norse Knighthood: The Impact of the Templars in Late Twelfth-Century Norway". In: Medieval History Writing and Crusading Ideology, ed. Tuomas M. S. Lehtonen et al., pp. 175-84. Helsinki: Finnish Literature Society, 2005.
• Harris, Richard L. "The Lion-Knight Legend in Iceland and the ValÞjófsstaðir Door." Viator 1 (1970): 125-46. https://doi.org/10.1484/J.VIATOR.2.301713
• Sif Rikhardsdottir. Medieval Translations and Cultural Discourse: The Movements of Texts in England, France and Scandinavia. Woodbridge: Boydell, 2012.

I think /u/y_sengaku gives solid answer to the question and would reiterate all his points. I would however like to add some additional points to contextualize some points about “Vikings” in the Middle Ages.

1. While the Viking Age refers to a mostly standard time period from the 8th-11th centuries. It is important to note that Scandinavian elites still went on “Viking raids” well into the 13th-14th centuries. The focus of these raids changed however from Western Europe to the Baltic region and they where often framed as crusades against the pagan peoples of the region.
2. In areas ruled by “Vikings” (more correctly Scandinavian elites/their dependents) in Western Europe the assimilation process began much earlier. Famously the Norman of norther France, but also rulers in modern day Netherlands. By assimilation here I mean the transformation to a more feudal society and adoption of mounted (knightly) combat.

The main point I am trying to make with these observations is that there is no exact point in place and time where Vikings became knights. Rather it is more useful tho think in terms of processes that led Scandinavian elites to adopt the trappings of Feudal rulership and in turn combat. These processes where different in different regions of Viking activity. Thus there are “Vikings” that fought like knights and knights/ crusaders that campaigned more like Vikings. One interesting example of this is the legends surrounding Henry bishop of Uppsala who was an English clergy who crusaded against the Finns in the mid 12th century and much of the iconography of him depicts the campaign using longships and being basically Vikings! With that in mind by the 13th century Scandinavia had completed their “western feudalism” transformation as suggested in the previous post. But they still sailed in ships to raid the Baltic coast much like the vikings of the early Middle Ages. But with a Crusading and chivalric trapping.