I don't think there is a universally accepted answer.
Some people are happy also to call the groups quartiles; and are thus explicitly or implicitly optimistic that any ambiguity will not bite, or at least can be clarified quickly in context, e.g. by inspection of some suitable table, graph and/or algebraic definition. There is a long history of such usages, sometimes distinguished by nuance, e.g. that the quartiles (values) may be called the lower quartile, median and upper quartile, while the quartiles (bins) may be called the first, second, third and fourth quartiles. (Such practice reminds me of those who want means to be population quantities and averages to be sample quantities, which to me has never seemed very convincing, not least because I really want the freedom to refer to sample means.)
Others would regard quarters as an alternative term.
The verbal alternatives all appear to buy greater precision by being more long-winded (and to some tastes more pedantic), say quartile-based bins, classes, groups or intervals.
In many ways the best solution is to avoid special words altogether: to be simply quantitative and talk about the first or lowest 25%, second 25%, and so on. [Grateful nod to @Glen_b for reminding me of this common practice.]
Yet another alternative is to avoid any such terminology altogether, but this is not always possible. There isn't a universal notation for quantiles either: for example, there are many idiosyncratic notations for median, but none seems even common.
The same terminological problem arises with any quantiles.
EDIT 8 Oct 2020 In almost five years since this answer I've seen the bins, classes or intervals delimited by quartiles (quantiles generally) often called by the same names. The ambiguity between intervals and the levels that delimit them is unfortunate, but seemingly here to stay. In practice the ambiguity does not bite hard. The natural selection at work is that longer-winded terminology such as quartile-based bins evidently seems too fussy to find favour.