As with any question about "medieval", it helps to specify when and where you're talking about. 10th century Scotland was pretty different from 13th century France, and 15th century Spain was something else again.

That said, I'm going to focus on France and England during the 13th to 15th centuries, because (1) you mentioned Game of Thrones, and GoT draws a lot of its inspiration from those places and times, and (2) those are probably closest to what most people think of when they think "medieval": castles, knights in armor, feudalism, crusades, and whatnot.

Okay so: the answer to your question is a resounding "sometimes". Sometimes lords lived on their estates, and sometimes they hung out at court. However, very broadly speaking, most lords spent more time at their estates. Most lords were at court at least sometimes, and some lords were at court most or all of the time, but court was not the default setting.

Let's narrow down and take a look at England in the late 1300s / early 1400s. At this time you had maybe 20 or 25 no-kidding capital-L Lords: three to six Dukes, and around 20 Earls. The numbers wobbled up and down as lines went extinct and new nobles were appointed or promoted, but those figures are good for the period ~1380 - 1430. So these 25 guys -- where were they spending their time?

Well, most of the time they were on their estates. In some cases this was driven by military necessity. For instance, the Earls of Northumberland -- the Percy family -- had their estates in the north, near the Scottish border. So they were usually Wardens of the Northern Marches, with military responsibility for border safety. Outright war with Scotland happened every generation or two, and in the meantime border raids were a constant background noise. So the border needed a strong hand, and the Earls of Northumberland were usually right there. In the southwest of England, the Earls of Devon were often made Admiral of the West and charged with suppressing piracy in the Irish Sea and the western approaches to the English Channel.

But even when there wasn't a military threat, lords had good reasons to spend time on their estates. Remember, an Earl or a Duke was expected to be a military leader -- meaning they had to have a bunch of knights and soldiers ready to fight. You wanted to see those guys regularly, to ensure their loyalty. Also, estates were almost always the noble's main source of revenue. Yes, if you had a trustworthy, competent person to run things for you, you could delegate the job and be an absentee landlord -- but stay away too long, and that person might slack off or grow corrupt and start skimming off the take.

But OTOH, yes some lords spent most or even all of their time at court. It helped if your estates were close to the capital. So, you'll see a lot more references to Earls of Kent or Surrey or Arundel hanging around London, since their main estates were just a day or two away.

Court had some obvious attractions. The King could appoint you to offices, which would allow you to increase your wealth, prestige, and/or power. When war came -- and war was never far away in those times -- you were more likely to get a plum military appointment. And you had the chance to keep an eye on all those other nobles at court, selfish scheming bastards that they were.

So for instance, picking a noble almost at random... Thomas Fitzalan, Earl of Arundel, who died in Henry V's siege of Harfleur. Arundel spent almost his entire adult life hanging around the courts of Henry IV and Henry V. When he wasn't actually at court, he was running errands for the King: suppressing a rebellion here, heading a royal diplomatic delegation over there. Arundel was a noble but he was also a courtier. Put him next to one of the rough-hewn border-fighting Earls of Northumberland and they'd look pretty different.

Now, obviously having a bunch of arrogant, powerful guys with local power bases and lots of armed men at their command was a recipe for various kinds of trouble. Noble rebellions, coups and attempted coups were a constant theme in the history, and nobles just ignoring royal commands was very common indeed. We all know about the Wars of the Roses, but those were just the most famous example. From Simon de Montfort to the French War of the Three Henries, uppity nobles were just a huge and constant PITA to the crown.

So, intelligent Kings tried to keep at least some of their nobles close at hand. In particular, medieval Kings tended to use nobles for what we'd consider Cabinet-equivalent job. After all, these guys were accustomed to lead, and they automatically got a certain amount of respect by reason of their social rank. Occasionally Kings would appoint commoners to high-level jobs in court; there were almost always a few of these guys around. But their position was always very fragile, and their noble colleagues tended to dislike and despise them. Later on, clever late medieval / early modern Kings like Louis XI of France and Henry VII of England would adopt a deliberate tactic of promoting commoners, as a first step towards developing a modern state administrative system. But that came later.

In the medieval period, the default expectation was that the most important posts at court and in government would be held by nobles. So, at least some nobles had to hang around the court to hold those posts.

So, if you visited a late medieval court in London or Paris, you'd think "wow, there's an Earl, there's a Duke, there are three more Earls... this place is full of nobles!" But then if you sat down and actually tallied up the names, you'd see that two-thirds or three-quarters of the nobility were still out in the countryside, sitting in their castles. Maybe they were peacefully administering their estates and waiting for the King to call them to service; maybe they were sharpening their swords and brooding on past wrongs. But they weren't at court.

(Very eventually, Louis XIV of France would hit on a brilliant solution to the problem of over-mighty noble subjects: build Versailles, a super-palace to hold a mega-Court, and turn all the nobles into courtiers, bickering over who got to hold the King's cup instead of lurking out in the provinces plotting. But that came much later, long after the end of medieval times.)

Additional wrinkle: some nobles held estates in widely separated areas. In particular, before the English lost the Hundred Years War, a lot of English nobles held estates in France. So those guys were often on the move from one area to another. Sometimes this meant they would, as it were, bounce through court on the way. A good example of this would be William Marshal, who was a triple threat -- estates in France, estates in England, spent a lot of time at the court of King John. But some other nobles were so busy trotting back and forth that they didn't have time to hang out in Paris or London.

Another wrinkle: war. Wars were pretty common through this period, and "Duke" and "Earl" were military titles as much as civilian. So when there was a war, the nobles tended to converge on wherever the battle was, which was usually (not always) where the King was. So, when Edward III or Henry V went to war in France, most of the country's nobility would join them. And when Richard III faced Henry VII-to-be at Bosworth Field, pretty much every important noble in England was there on one side or the other. In the case of peripatetic warrior kings like Edward I, Henry V, or (especially) Richard the Lion-Hearted, this meant that nobles might be away from their estates, warring alongside the King, for months and years at a time.

Is this the sort of thing you're looking for?