In one of the scenarios proposed for explaining away ZF-COSMOS-20115, a
merger of two galaxies is proposed. How much time do those mergers take
in years?

A hint can be found in the work of Jennifer Lotz, of the Hubble space
telescope team.

http://hubblesite.org/news_release/news/2011-30

Scientific paper:
arXiv:1108.2508v1 [astro-ph.CO] 11 Aug 2011

She arrives at time scales of a BILLION or more years for a full merger.

So, to build the quasar by a merge of two big galaxies we need around a
billion years, and in that time the time needed to heat the gas in the
galaxy is not even considered!