Yes. The Quaternary Period is part of the late Cenozoic.
Humans currently live in the Holocene stage. This is part of the Quaternary period, and the Quaternary is a part of the Cenozoic era. The Quaternary began 2.588 million years ago.
Cheetahs are alive today, which puts them in the Quaternary Period in the Cenozoic Era. The oldest Cheetah fossils date to the late Neogene Period, which is also part of the Cenozoic.
Mammoths lived during the Pleistocene series, which is part of the Quaternary system, and that is part of the Cenozoic Era.
The earliest hominids evolved in the late Neogene period, about 3 million years ago. Presently, we live in the early Quaternary period. Both of these periods are part of the Cenozoic era.
The Neogene Period is not a part of the Quaternary Period. The Quaternary Period occurred right after the Neogene Period. The Neogene is divided into two, Miocene and the Pliocene epochs.
The Cenozoic is a major geologic era starting at 65 million years ago and encompassing the present. The Pleistocene Epoch is a much smaller division of time that is part of the Quaternary Period of the Cenozoic Era, from 1.8 million years ago to 10,000 years ago.
The wooly mammoth lived in the Cenozoic era, the Tertiary period, Pleistocene epoch, and the Calabrian age.
Saber toothed cats died out 10,000 years ago. That was at the end of the Pleistocene, the first series of the Quaternary. The Quaternary is the current period, and it is part of the Cenozoic era.
The Quaternary is the most recent geological time period, which started 1.67 to 2.4 million years ago depending on the area studied. The Quaternary is part of the Cenozoic ("new life") Era, along with the Tertiary Period. The Quaternary Period includes recent time (the last 11,000 years ago, since the last ice age) which is called the Holocene and the Pleistocene Epoch. There is currently much debate as to how long the Quaternary period is. There are considerations into re-classifying the Quaternary as it is "technically" too short to be considered a geological age. But this is a matter of much conjecture at the moment.
The saber toothed cat subfamily, Machairodontinae, first evolved about 23 million years ago and became extinct 10,000 years ago. That means that they first appeared during the late Paleogene period, and they lived through the Neogene and the early Quaternary periods. All three of these periods are parts of the Cenozoic era.
Cenozoic is not a major period of the geologic time scale. The scale is broken down into eras. Cenozoic, Mesozoic and Paleozoic are all part of the Phanerozic era.
The period after the Cretaceous is the Paleogene period, which is part of the Cenozoic era. It includes the Paleocene, Eocene, and Oligocene epochs.