The short version is yes in a way, but not in the exact manner depicted by the show. There is quite clear evidence that suggests gifts were inextricably tied with loyalty and service by a lord's followers.

One of the most important duties of the lord in Germanic societies, including the Norse, was ensuring that his followers were rewarded for their loyalty and service with gifts, often of jewelry.

Much of our evidence, particularly in literary contexts, comes from other societies besides the Norse, largely due to an extreme lack of firsthand sources from Scandinavia before conversion.

Anglo-Saxon writings in particular are absolutely full of references to leaders who are generous with their followers, particularly with items such as rings. (as a quick aside: Tolkien later drew on this tradition in crafting his own works)

Beowulf mentions the practice of ring giving and the expected generosity of good rulers in passages such as

Heorot he named it Who wide-reaching word-sway wielded ’mong earlmen. His promise he brake not, rings he lavished, Treasure at banquet

Now, as to whether oaths would be sworn on these sorts of jewelry in particular. Oaths and vows are sworn quite often in Anglo-Saxon and Norse writings, as well as in legal codes, and they were usually accompanied by some sort of object to be sworn on, usually an object that held deep religious significance.

There isn't a clear preference for objects that were given by the lord to his followers according to Gregory Liang's "Bound by Words: Oath-taking and Oath-breaking in Medieval lceland and Anglo-Saxon England". Instead the gifts and rewards for service were the expectation of the followers on the part of the lord, swearing over these objects was unnecessary as their transfer was itself a form of oath.

Sources: Gregory Liang's "Bound by Words: Oath-taking and Oath-breaking in Medieval lceland and Anglo-Saxon England"

This is a phD dissertation for a degree in English I believe, but the sources and work done are easily applicable here, especially since so much of what is known about these oaths is literary in origin.

Beowulf, trans by Lesslie Hall