While the word trivia has come to mean "useless knowledge", its original meaning was nearly the opposite. A similar word is elementary, which has come to mean "easy" rather than fundamental. Euclid's Elements is not an easy book to understand, and Jerrold Marsden's Elementary Classical Analysis is a book that someone who just finished a year of university calculus with an A grade would have difficulty understanding fully (which does not mean that it is not a great book).
The opposite of trivial is non-trivial, a word which can have savagely ironic meaning. I heard a lecture by the great number theorist Ken Ribet, in which he sketched Andrew Wiles' proof of Fermat's Last Theorem. Fermat first stated the conjecture, without proof, in 1637. Fermat claimed "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain." For more than 350 years, hundreds of mathematicians scrambled to find any proof, to no avail. Finally in 1993, Andrew Wiles submitted a proof about 150 pages long, in which several new branches of mathematics had been invented just to prove this result. The proof was flawed, and it took Wiles and a graduate student two years to fix the flaws in his theory of Galois Representations, which trimmed the proof down to a manageable 100 pages or so. In 1995, a team of experts declared that this new proof was correct. At his lecture, Ken Ribet wrote the proposition of Fermat's Last Theorem on a blackboard, and then said, "The proof of this result is non-trivial."
But nobody uses the adjective quadrivial. The Quadrivium consists of Arithmetic, Geometry, Music and Astronomy. Arithmetic is the study of number. Geometry is the study of number in space. Music is the study of number in time. Astronomy is the study of number in space and time. These subjects were considered to be more difficult than the subjects in the Trivium.
By Arithmetic, the modern subject of Number Theory would be more appropriate. This includes the theory of prime numbers, factoring, divisibility, continued fractions (a fun subject hardly anyone studies anymore), Diophantine equations (or algebra restricted to whole numbers). The modern subject includes a lot of cryptography. If you use a credit card on the internet, chances are that it is being encrypted with a public key encryption as well as symmetric encryption algorithms, which involve factoring massively large prime numbers. Not many masons still use the Pig-Pen Cipher, but any simple substitution cipher can be cracked in milliseconds by a decent computer. Because of the easy availability of computers millions of times more powerful than the computers that fly the Space Shuttle, almost anyone has access to cryptography so powerful that no government or military agency can crack it within the time frame of human lives. Depending on your temperament, that's either thrilling or terrifying.
Geometry is what is today called Synthetic Geometry, which starts with Euclid's Elements, but continues into a study of Conic Sections. It could be argued that, after Newton, this would include Analytic Geometry and the Calculus, especially after Descartes. Nobody today studies Conic Sections the way the Greeks taught the subject, and the way that it was studied up until Kepler. A little bit of algebra makes a very difficult subject much simpler, and most high school students study Conic Sections using algebra, rather than straight-edge and compass.
Music would today be called Music Theory, especially the theory of harmony and harmonics. Pythagoras' treatment of music was mathematical in nature, showing the ratios of the lengths of strings of various musical intervals. A taut string one half the length of a given taut string will sound at one octave higher in pitch. Considering the original string to be the tonic, a string 3/4 the length will sound at a major fourth, and one 2/3 as long will sound at a major fifth. One 8/9 the length will be a whole step higher. The classical theory of music uses these ratios to derive scales, modes, harmonies, counterpoint, and other ideas in music theory.
Astronomy would today probably include Physics. We know so much more than the ancients did about this subject that the modern subject is totally different from the classical subject. The ancients believed that the sun orbited the earth, which seems pretty obvious except that it is not true. Explaining why this is not true is not easy (informal exercise for the reader: come up with a convincing argument why the earth actually orbits the sun, rather than the reverse. It is much harder than you think!). Because there's no immediate reason or evidence for the actual state of affairs, the ancients kept the geocentric model. To improve upon it without giving up the basic premise, they had to come up with little mini-corrections, called epicycles, to get the theory to match the data. Mercury, for example, has an orbit that requires Einstein's General Theory of Relativity to plot accurately. From the point of view of all the planets and the sun orbiting the earth, it appears to stop in its orbit, and go backwards from time to time, then stop again, and go forwards again. The culmination of ancient knowledge about Astronomy is the Almagest of Claudius Ptolemy, which is Arabic for the The Great Book (Al- usually means "the" in Arabic). It fails in the accuracy department, but is very elegant. The Catholic Church fused the Almagest into their cosmology in medieval times, and both Saint Thomas Aquinas and Dante treat this theory as a general fact. Copernicus discovered that the math gets much easier if you assume that the earth and other planets orbit the sun. He was careful to warn his readers that his model did not describe reality, but was a convenient fiction to make the calculations easier. When Galileo dared to assert that the solar system actually was heliocentric, the Holy Inquisition invited him to tour their torture chambers and inspect the instruments there.
Masons are told that of the Quadrivium, we should have an especial love for Geometry ("or Masonry" the Preston-Webb Monitor cheekily asserts). As one of the few Freemasons I know who can take a compass and straight-edge and use them to construct and prove the 47th Problem of Euclid, I'm aware that not all masons take this admonition to heart. It is worth pointing out that the future President Garfield, while Chaplain of Garrettsville Lodge #246 in Ohio, invented a new proof of the Pythagorean Theorem.
The Trivium and the Quadrivium together comprise the Seven Liberal Arts. Today we use the term Liberal Arts to mean the Humanities. Ask a Liberal Arts major to reduce a rational number to a continued fraction, to inscribe an ellipse in a parallelogram, to compose a melody in G Mixolydian, or to predict the next perihelion of Venus, and you might get smacked for your trouble.
I'm not as firmly convinced that the study of the Quadrivium is as central to Freemasonry as the study of the Trivium. While masons had a large influence on the creation of the Royal Society, and much of the Scientific Revolution, there's not much of the Quadrivium that is essential for every modern mason to learn. Certain numbers show up again and again in our rituals, and an understanding of these numbers and their relationships is very illuminating. I personally believe a mason should know the proof of the 47th Problem of Euclid, and know why the dimensions of the lodge room are what they are, but one can be a fine mason without that knowledge. At least one person in a lodge should know how to play the organ or a similar instrument, because every lodge needs music. While our ritual mentions a few heavenly bodies, I'm not sure a mason needs to know more about them than a basic public school education teaches. If anything, the symbolic meaning of these things is more important than a scientific understanding of them.
Every mason must be a friend of science, an ally of true knowledge. A mason sees no conflict between science and religion. We understand that the Grand Architect of the Universe leaves information for us to perceive for our own edification, and if a passage in a particular Volume of Sacred Law contradicts what the GAOTU is showing us, we know how to sort out these little discrepancies without insulting either source of knowledge. The value in studying the Quadrivium comes from a improved epistemology: by learning a myriad of things we can know, we learn how it is we actually know what we know.