Earlier quoted context omitted.
ISO 8601-2:2019 defines quarters in section 4.8, "Sub-year groupings". Your example of "2021-Q1" can be represented as "2021-33".
Unfortunately the document is not freely available. Can you elaborate how it works?
Representations of seasons, quarters, etc. is one of many types of extensions included in Part 2.
These extensions are always used in "YYYY-MM" format (and always include the "-"). For the "MM" field, you use the numbers 21-41 inclusive. The numbers represent, in order, spring, summer, autumn, and winter, in 3 ways: independent of hemisphere (21-24), northern hemisphere specifically (25-28), southern hemisphere specifically (29-32); quarters 1-4 (33-36), quadrimesters 1-3 (37-39), and finally, semestrals 1-2 (40-41).