Does 2 merging black holes necessarily make a quasar?
Basically no. While the merging of 2 black holes is a very interesting event, a quasar is what you get when 1 very large black hole eats a whole bunch of matter and the light from the quasar comes from the intense heat and interactions from that tightly bound, rapidly spiraling and very excited matter.
Quasar's were probably most common when galaxies are young but there are a few more recent ones (see examples in comments). Quasar
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If the size is smaller than what would be considered as a supermassive
black hole would it not be a quasar?
Almost all large galaxies have a super-massive black hole in their center. Source. The sizes vary with the size of the galaxy. Quasars are much more rare, only in a few galaxies.
As for the specific size of black hole that can form a quasar, credit to Rob Jeffries below.