Quasars are known to contain supermassive Black Holes at their cores. So does that mean that the number of quasars is equal to that of the SMBHs out there?
It means that the number of quasars must be less than or equal to the number of SMBH out there. It's believed that supermassive black holes are found at the center of most galaxies and the SMBH mass correlates with the velocity distribution of stars in the galaxy.
However, quasars only form in particularly large and active galaxies. Or perhaps particularly massive galaxies can only form around and be supported by particularly massive black holes? Regardless, based on this, while every quasar must possess a supermassive black hole, not every supermassive black hole is part of a system that contains a quasar.
Very good question. The number of quasars must be less than the number of SMBHs, since many galaxies, such as our own, contain SMBHs at their core (Sagittarius A*) and they are not classified as quasars (i.e., some galaxies are quiescent for whatever reason). Quasars represent an ultra-luminous active phase of gas accretion onto the SMBH. Such larger luminosities is believed to be caused by intense gas accretion triggered by major massive scale mergers between galaxies.
As such, Quasars are short lived events, and the SMBHs outlive the quasar (the lifetime of a quasar is of the order of 10$^6$-10$^9$ yr whereas the lifetime of a SMBH is much greater than the Hubble time). Hence, once the gas is all but consumed by such intense accretion the Quasar will slowly become quiescent. Hence why many galaxies contain SMBHs at their cores, but are no longer active.
Not really. Quasars do indeed need a SMBH to be powered, and this is a necessary condition.
But, Quasar phase only lasts for tens millions to few billions years. This means that, in a more recent universe, Quasars are turned off but the host galaxy... is still there. And the SMBH also do.
This is exactly as our Galaxy case ($Sgr A^*$), where the evidences for the presence of a SMBH are among the strongest ones, but still no nuclear activity is present.
However, there are tons of study to infer the proper amount of AGNs in the whole Universe by using luminosity functions.