I could not find a satisfying definition of this term from anywhere. Also, Can you please give some introductory level references to understand quantum gas?
To put it simply, a quantum gas (or quantum fluid) is a system made of indistinguishable particles, as opposed to a classical gas where you assume that you can differentiate one component from other (that means that in a classical gas you could label every single constituent of the gas theoretically, and always know its position and momentum, whereas for a quantum gas you cannot know such information for each particle individually).
Very generally, fluids exhibit a quantum behaviour at low temperatures and/or high densities and the classical gas model is a good approximation for high temperatures and/or low density. However, don't get confused: there are no classical particles in nature and the classical gas is nothing but a simplification that under some circumstances can be used to describe reality.
Depending on the wavefunction simmetry of the particles that constitute the gas, the gas follows a different statistic, namely Bose-Einstein statistics for bosons and Fermi-Dirac for fermions.
From a macroscopic point of view, we usually call quantum fluids to those that exhibit some kind of "quantum" macroscopic behaviour such as superfluidity or superconductivity.
To start studying quantum gases I think that it would be necessary to have some background in statistical mechanics and also some ideas of quantum mechanics.
For the statistical mechanics part:
H.B. Callen. Thermodynamics and an introduction to Thermostatistics.
S. R. A. Salinas. Introduction to Statistical Physics.
D. J. Amit, Y. Verbin, R. Tzafriri. Statistical physics. An introductory course
And maybe more advanced options are
R. Balian. From microphysics to macrophysics.
M. Kardar. Statistical physics of particles.
And for the quantum mechanics part I'd reccomend you
Cohen-Tannoudji C., Diu B., Laloe F. Quantum mechanics, vol. 1 and vol. 2