The terms Tertiary and Quaternary are no longer part of the recommended nomenclature for naming geologic time. If you want to know why not, please read below.
Way back in the history of geology (circa. 1759), Giovanni Arduino proposed that all of geologic time should be broken up into three great orders: the Primary, Secondary, and Tertiary. The Primary and Secondary were rough analogs of the Paleozoic and Mesozoic and these more modern terms have taken their place. By contrast, the term "Tertiary" has survived and is still in common use today. In 1829, Jules Desnoyers added the term "Quaternary" to describe a sequence of rocks apparently younger than what were then regarded as the youngest Tertiary deposits. Since his time, the Quaternary has more generally been used to describe the recent period of cycling glacial and interglacial climates. However, because of the historically ambiguous way in which the boundary between the Tertiary and Quaternary has been defined, these terms have fallen out of favor for formal scientific use, and the International Commission on Stratigraphy no longer endorses them. Instead the terms Paleogene and Neogene, in their modern form, are the preferred way to subdivide the Cenozoic Era.
While it is generally accepted that the Tertiary began at the end of the Cretaceous and that the Quaternary continues until the modern day, there has never been a good agreement on where the Tertiary ends and the Quaternary begins. Essentially three leading historical definitions have been used and still occur with varying frequency. The first is to associate the boundary with the start of the cycles of glacials and interglacials that characterize recent climate (~3.0 Ma). This climatological definition is quite popular but unfortunately it does not fit easily into the system of geologic stages that have been defined in other ways. In particular, the Pleistocene and Pliocene Epochs (first described by Lyell in 1839) are separated by a boundary at ~1.8 Ma. By usual convention high level divisions of geologic time (e.g. Periods & Eras) must also coincide with divisions at the stage and epoch level. This has resulted in some authors choosing 1.8 Ma as the division between the Quaternary and the Tertiary. Still others endorse something of a compromise position. They ignore the timing of the Epoch boundaries, but fix the Tertiary-Quaternary boundary at the start of either the Gelasian (~2.6 Ma) or Piacenzian (~3.6 Ma) stages, as such choices are nearer to the climate transition. Ultimately though, none of these ad hoc definitions are satisfactory since they lack the intuitive sense of meaning usually attached to high level divisions of geologic time.
Adding further confusion, the terms "Paleogene" and "Neogene" have been widely used to describe Periods within the Tertiary since their introduction by Hornes in 1853. Because of this, the Tertiary and Quaternary are often termed "Sub-eras" within the Cenozoic Era rather than Periods themselves. But if the Quaternary is not a Period, then what Period(s) occur within it? Some authors have endorsed the use of a single Period within the Quaternary often referred to as the Pleistogene. Still others have chosen to ignore the Tertiary-Quaternary boundary and extend the Neogene all the way to the present.
To bring clarity to these issues, the International Commission on Stratigraphy ultimately decided that the simplest solution was to do away with the terms Tertiary and Quaternary. So now the Cenozoic will formally be divided into only the Paleogene and Neogene periods, with the Neogene having been designated as continuing until the present. This avoids much of the historical ambiguity and confusion, and provides a clear roadmap for scientific use. Unfortunately, this also means losing terminology with nearly 250 years of history. Somehow, it doesn't seem likely that there will be a rush to abandon the term "Tertiary" in popular use. For example, it will probably be a long time (if ever) before the extinction of the dinosaurs at the Cretaceous-Tertiary (or KT) boundary becomes associated with the "Cretaceous-Paleogene (or KPg) boundary".