Vikings is a great watch, very entertaining, but you're right to question everything - it contains many serious factual errors, mostly pertaining to what Vikings of "the time" knew existed and the places they had visited.

For example, the idea of Ragnar sailing west to find new land (the place we call the British Isles) is several hundred years too late. People from the northern mainland and southern Scandinavia had been going there to raid and/or settle for at least a few hundred years. The idea of Bjorn exploring for the "sea with no tides" (the Mediterranean) is also bunkum, there were well established trade routes between the northern coasts and the Med that ran around the mainland coast and across the interior.

The route, or rather the extended network of routes that we know as "The Silk Road" definitely existed at that time and had done so for nearly 1,000 years, but it wasn't called by that name until about 150 years ago when the term was coined by Robert Mack in 1874 in "The Importance of the Black Sea for World Trade".

There are enough written records about the Vikings (or Vikingrs, Wykers, Village-Makers) for us to have a good idea of how they conducted themselves. We know that they were very good at four things in particular: sailing, farming, trading, and hitting things repeatedly until they were dead.

Before their Norman era their culture was largely verbal so our particular inferences about their extensive geographical knowledge and exploitation of far-flung trade routes have to be based on the archaeology of their settlements, hoards and burial sites. We know from numerous finds that they were able to get their hands on silks, rare gems, and many goods that were only available in the Far East. That's not to say that all Vikings were travelling the "Silk Road" but they clearly had effective access to the supply chains. This trade access was more often peaceful than violent, we have good evidence that Vikingrs liked to settle alongside existing towns and integrate with local trade.

In summary: There's no evidence that the name "Silk Road" existed before the 1870s although the routes came into existence in around 200BC. The Vikings certainly knew about the trade connections and used them, although there's little evidence of them travelling the routes themselves in any significant numbers.

Sources:

Cultures in Contact - Scandinavian Settlement in England in the 9th and 10th Centuries, Hadley D, Richards J, 2000

The Winter Camp of the Great Viking Army - AD872-3 - Torksey, Hadley D, Richards J, 2016

The Importance of the Black Sea for World Trade, Mack R, 1874

Isotopes and Human Burials at Viking Age Birka and and the Malaren Region of East Central Sweden, Price T, Arcini C, Gustin I, Drenzel L, 2018

Silk Trade to Scandinavia in the Viking Age, Vedeler M, 2014