Do I need an I-Q (in-phase/quadrature) mapper in Simulink with the QAM modulator? Would it be correct if I use the Random Integer Generator block with a certain $M$-ary number (say $M=16$) followed directly by the General QAM Modulator Baseband block?
I saw an example where they use an IQ mapper between the two blocks. Is it necessary? 


I assumed the QAM block already does the mapping (inherently) because when I read the document it was specified that the input of the QAM modulator block should be an integer between $0$ and $M-1$, with no mention of the IQ  mapping.
You don't need the mapper. The Simulink QAM modulator lets you specify a vector of constellation points. The input should be an integer (or vector of integers) that select elements from the constellation points.

By default the block specifies a 8-PSK constellation. If you want something different, you need to provide the vector of constellation points.