I have read in here: https://math.stackexchange.com/a/2899795/588260 that $\mathbb{Q}$ is 'larger' than $\mathbb{Z}$.
I assume it has to do with transfinities in which the one of $\mathbb{Q}$ has a relation of order so that it above the one of $\mathbb{Z}$, but I haven't read about that yet. I am just curious if someone can explain to me this concept, I know very basics of transfinity, like what $\aleph _0$ means, but not that much.
You've misunderstood your link. The statement
Is an example of a misconception — a popularly held belief that is not actually true.
The relevant technical notion here is cardinality. The phrase "X is a larger set than Y" is a colloquialism for "The cardinality of $X$ is greater than the cardinality of $Y$".
Showing there exists a bijection between $\mathbb{Z}$ and $\mathbb{Q}$ proves that the proposition "The cardinality of $\mathbb{Q}$ is greater than the cardinality of $\mathbb{Z}$" is false.
Incidentally, I don't think I've every heard the term "transfinity" outside of some... eccentric articles.
The term "transfinite" does appear in mathematics, but almost always in a very niche usage: specifically referring to sequences (and things related to sequences) whose index set continues beyond the natural numbers. For example, the sequence consisting of all natural numbers, and then followed by the ordinal number $\omega$ would be an example of a "transfinite sequence".
Instead of the word "transfinity", I think the words you're looking for are along the lines of "infinite set" or "infinite cardinality".
Also note that in mathematics, "infinity" is usually used as a proper noun — e.g. to refer to the extended real number $+\infty$. The adjectival form "infinite" is more common. E.g. one says "an infinite set" rather than "an infinity", or "you have infinitely many things" rather than "you have an infinity of things".