There is a relationship, though how deep this is depends on various factors. You are right in your hunch that Swabia (= Schwaben) and Suevia are connected etymologically.

Ethnic labels are a very difficult thing to handle, now as in antiquity, and it is, especially in antiquity, often unclear whether such an ethnic label is an external adscription that got adopted to some degree or another by those who were labeled thus, or if it is an internal label that got adopted by the outside world. Ancient ethno- and geographers segmented the people of Northern Europe into Celts (Keltike) in the northwest and Scyths (Skythike) in the northeast, thus the Cimbri and Teutones were held to be 'Celts' or 'Celto-Scyths' at first. Only with the increasing Roman northward contacts and Caesars occupation of Gaul it became clear that there was another group of people that didn't seem to speak a Celtic language and who were clearly no Scyths, whom Caesar labeled 'Germans', and thus created a new ethnogeographic category to order the space between Rhine and Eastern Europe. Most prominent among them at first were the Suebi and their leader Ariovist, whose intervention in Gaul formed a convenient casus belli for Caesar.

The term Suebi is a bit problematic, since it is used by mediterranean authors as a kind of umbrella term for a confederation or group of several German sub-groups, commonly labelled Elbgermanen, because of their core area in Northern Germany. The term is used by Tacitus that way, and he includes f.e. the Semnones and Langobards, the Hermundures, Markomanns and Quades among the 'suebic' groups, he also called the Baltic Sea mare suebicum, the Suebian Sea. The geographer Claudius Ptolemaios also includes a number of German tribes under that label. In any case, the Suevi or Suebi were in contact with the Roman world in the late Republic and the Augustean age, and reached as far southwest as Switzerland and France. After the defeat of Ariovist, it seems that groups of foederati or 'allies' who were called suebi settled (maybe forced, resettled by Rome) in the areas now under direct or indirect Roman control, along the right bank of the Rhine south of the river Main. They were highly favoured as warriors, and some of them served in the imperial elite cavalry forces. One group of them, settling in the modern area of Heidelberg, Mannheim and the lower Neckar was called Suebi Nicrenses, the 'Neckar Swabians'. They, like other communities, were constituted into a civitas, the lower tier Roman municipalities, often designed along ethnic lines to keep the groups under close Roman control. Thus, they were often named after the people they included, and the Neckar Swabians are the only 'Swabians' by that name living in the Roman Empire. The area they lived in, the lower Neckar valley and the fertile plains of the Rhine river valley around their municipal capital of Lopodunum (modern Ladenburg) is not directly identical with modern 'Swabia', but overlaps with it. Other areas of the province were incorporated under Roman rule as well, but the other local municipalities of the area of modern Swabia are either not known by Name or did not carry any ethnic name. In any case, it seems that the Roman Swabians acculturated themselves quite well in the Roman Empire, and they were succesful warriors, tradesmen and farmers. This member of the Pretorian Guard is called a Suebus Necresis, a Neckar Swabian.

Roman control of modern Swabia was not long lasting, though, and in around 260 AD Roman control over the area was ended by the invasion of the Germanic Alemanni, who occupied the right bank of the Rhine from then on. The area was then called Alemannia. The original 'Swabians' and 'Suebi' disappeared, or went to Spain, or got absorbed into the newly arrived Alemannians. It's not precisely clear how the area got its modern name of 'Swabia/Schwaben', but it's clear that 'Alemannia' became superseded by it. One factor must have been that 'the Alemanni' didn't exist as a somewhat ćlearly defined group until that southwestern push by several groups that were now living together, who called themselves 'Alemanni' (literally 'all men'). The groups of warriors that became the 'Alemanni' came from mainly northeastern Germania, across the Elbe, from those areas then settled by Slavs; and some of them must have included people from the 'traditional' suebic areas and groups on the Elbe, such as Semnones or Iuthunges (who invaded Italy in the late third century) and thus brought a suebic element that could be used as a core for a crystallization of tradition, much as the Franks in middle Germany came to call themselves 'Thuringians'.

Jordanes, writing in the 6th century, still used 'Suevi/Suavi' and 'Alamanni' as separate terms, but saw them as close allies and neighbours:

With the Suavi there were present the Alamanni, then their confederates, who also ruled the Alpine heights, whence several streams flow into the Danube, pouring in with a great rushing sound. Into a place thus fortified King Thiudimer led his army in the winter-time and conquered, plundered and almost subdued the race of the Suavi as well as the Alamanni, who were mutually banded together.

The Suavi mentioned by Jordanes are probably Quadians or 'Donauschwaben', 'Danubian Suebians', who lived along the Danube and played a big part in the so-called 'second' ethnogenesis of the Alemanni in the 6th centuries, from which time on they also begun to be called 'Swabians' and the area of Alemannia known also as 'Swabia'. I can't go into detail about how those labels were used in medieval times, because there still is some ambiguity about how the area and the people living there are called (and even today, the area is called 'Schwaben', the people as well, but the dialect 'alemannic'), not to mention the gaping hole that is my knowledge of the time between 600 and 1800. But I hope I was able to show that yes, there is a very close connection, even though it's not that easy to determine where and how exactly that came to be - but the easy answer that it's called Swabia because germanic people who called themselves 'Swabians' or variants thereof did live there doesn't seem to be too far off the truth.

*an admixture of