An existing question asked can Q destroy the universe, and while this question is not the same, it includes some of the supporting details for an answer:
Could the Q destroy the universe?
Yes, they could.
As shown in various episodes, the Q have the power to create
new dimensions, alter the very fabric of reality in any way they
see fit and they can even hurt and kill themselves and each
other and bring untold destruction on their own continuum.
In Qpid (TNG S04E20), Q creates a temporary pocket
dimension based on the story on Robin Hood, with the officers
of the Enterprise playing Robin Hood and his Merry Men,
much to the dismay of Worf. Q is able to create this dimension,
give it an end time and even prevent himself from being able to
interfere with it or know what's going to happen.
Because a pocket universe is still a universe, the answer is yes, Q can create a universe.
With their ability to create a universe established, the ability to introduce randomness of its generation and alien population can be surmised. Q powers increase the more act in unison; if one Q couldn’t do it, then X number of Q necessary for the particular plot likely could.