There is a link, although it is of almost purely linguistic persuasion.

Teutons were a tribe or nation of Germanic people, although some historian argue that they could also belong to Celtic group, as even their ethnonym suggests Celtic origin (cf. 'tuatha'), inhabiting territory of modern Germany (south of Jutland, in the vicity of Elbe river). They went into prominence in the late 2nd century BC, when, together with Cymbres and Ambrones defeated Roman army and few years later they were subsequently defeated. This war was one of the first contacts between Romans and Germanic tribes and made it clear that strengthening of the northeastern border of the Republic is necessary. Then, the name could have been conflated with 'theodisc' (a vernacular form of German, group of languages spoken in the Eastern part of Charlemagne's lands) somewhere in late 8th century, leading to the term 'teutonicus' replacing 'germanicus' in Middle Age Latin.

In Medieval Latin the term 'Teuton' was a contemporary equivalent to modern 'German'. For example, Ordericus Vitalis, when writing about German duke Balderic in early 12th century calls him 'Balderic Teutonicus'. The 'Continuatio Prædictorum Vindobonensium', in a passus referring to the murder of Gertrude of Merania (daughter of an Bavarian noble and thus considered German) notes "patriarche Aquilegensi uxorem Bantzi procaverat, qui teutonice Prenger vocatur' while 'Annales Gotwicenses' states 'Ungarorum meliores armata et violenta manu contracta odium Teutonicorum reginam occidunt'. Similarly, Thietmar's Chronicle (Thietmari Merseburgensis episcopi Chronicon) states in the third paragraph 'Isque a patre suo in provinciam, quam nos teutonice Delemenci vocamu, Sclavi autem Glomaci appelant[...]', while a ducal privilege issued by the Casimir, duke of Opole to Laurentius, bishop of Wrocław in 1222 states '[...]et concedo libertatem plenariam in territorio beati Johannis in Uiasd ad locandum Teutonicos vel alios hospites in iure Teutonico vel alio modo[...]' (and give plenipotency to settle Germans and other guests in the St. John's territory in Ujazd, according to German law or other laws' [in Polish, the term 'German law' was used in reference to Magdeburg rights]). Later example of such use of the term can be a Latin-Czech-German dictionary published in 1513 in Vienna under the title 'Dictionarius trium linguaru latine, teutonice, boemice potiora vocabula continens peregrinantibus apprime utilis'. Also, 'Chronicon Hiersolymitanum', a collection of sources on crusades compiled by Reiner Reineck and published in 1584 contains passages as 'duos dies Teutonicorum victoriae ad urben Nicaeam' (two days from German victory at Nicaea), 'Dux Godefridus et Teutonicorum comitatus obsidione premebat' (Duke Godfrey [of Bouillon] with German retainers pressed with a siege), 'viri de gente Teutonicorum ex Regnesburg' (men of German nation [lit. family] from Regensburg).

So how all of this relate to the Teutonic Order? Now it is simple. The Order was initially a community created by German knights to protect and assist a field hospital founded by crusaders and pilgrims from Lübeck and Bremen during the Third Crusade, initially called simply 'German Hospital' and later 'German Hospital of St. Mary in Jerusalem'. Pope Innocent III transformed this community into a fully fledged Knightly Order that started using using Templar rules and adopted the name of 'Order of Brothers from the German Hospital of St. Mary in Jerusalem' (ger. Orden der Brüder vom Deutschen Hospital Sankt Mariens in Jerusalem). Latin was the official language of the Church, this was automatically rendered into Latin as 'Ordo fratrum domus hospitalis Sanctae Mariae Teutonicorum Ierosolimitanorum', and as the names of the orders and ecclesiastical organisations were often shortened for convenience (e.g. Poor Fellow-Soldiers of Christ and of the Temple of Solomon became known as 'Knights of the Solomon Temple' or simply 'Knights Templar'), the newly founded order was commonly called 'Ordo Teutonicus' that was then translated into English as 'Teutonic Order'. It should be noted that this name is generally used only in English and Eastern Slavic languages (e.g. rus. Тевтонский орден), while Germans themselves called themselves called the order 'German Order' (Deutscheorden, later also 'Deutscheritterorden' and 'Deutscheherrenorder'), what is also reflected in the names used by Hungarians (Német Lovagrend) and Lithuanians (Vokiečių ordinas). Curiously, Poles, were initially using the name 'German Order' (pol. Zakon Niemiecki) but quickly adopted a local name that translates roughly as 'Crossers' (pol. 'Krzyżacy') complete with Latin counterpart ('Cruciferozi'), as evidenced by e.g.'Bandera Prutenorum' (Prussian Banner), a book written in 1448 by Jan Dlugosz (Ioannes Dlugosius) and describing in detail all the enemy banners captured by Poles and Lithuanians after the Battle of Tannenberg.

So, summing it up, the name 'Teutonic Order' is a result of an English translation of the Latin equivalent of the original name of German Order, resulting from the tradition of using the name 'Teuton' to denote inhabitants of Germany, hailing back to the Roman texts of 2nd century BC.