Is there a standard date format that expresses Quarters?

When it comes to standards, we can distinguish two types, de jure and de facto. The first type is the one defined by some established authority, whereas the second one is the type that emerges in practice over time, often informally.

While the predominant standard in date and time formats is ISO-8601, currently it does not explicitly specify a format for quarters. Something curiously similar, admittedly little to do with your issue, is the week-of-the-year part of the specification, allowing formats along the lines of 2018-W30. Actually, ISO allows for variations, thus giving the implementation some freedom.

Putting ISO aside, due to its dominant status, it is very probable that any standards you encounter are just local or partial. You can see this in the following explanation in Investopedia:

Not all companies use the uniform quarter standard. For example, Wal-Mart Stores’ first quarter is February, March, and April; Apple Inc’s Q1 is October, November, and December; Microsoft Corporation’s Q1 is July, August, and September; etc. In addition, certain governments use different quarter systems. For example, the first quarter of the United States federal government’s fiscal year is October, November and December, Q2 is January, February and March, Q3 is April, May and June, and Q4 is July, August and September. State governments, also, may have their own fiscal calendars.

Whereas it talks about processes, and not writing formats, it is indicative of the variance you might encounter in your search. At the end, I’d suggest that using an intuitive approach as the one you suggested, or constructing yourself a format along the lines of ISO’s week formats (thus 2018-Q2 or 2018Q2), might be your best hope to be aligned to some standard. However, notice @maaartinus’ comment demonstrating that the ISO-influenced variation is more workable when it comes to sorting, even when the data is treated as strings.

When I personally had to report aggregated data, I also used two values (one for years and one for quarters), because this allowed for more flexible processing (e.g. filtering) after that. However, as indicated in the answer by @maaartinus, this is easily achievable as an expression, rather than having it stored as separate columns.