Can Random Number Generator of Fortran 90 be trusted for Monte Carlo Integration?

The particular random number generator used depends on the compiler. Perhaps documented, perhaps not. Perhaps subject to change. For high quality work, I would use library / source code from elsewhere. A webpage with Fortran implementations of the Mersenne Twister: http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/VERSIONS/FORTRAN/fortran.html. I have used http://theo.phys.sci.hiroshima-u.ac.jp/~ishikawa/PRNG/mt_stream_en.html and verified against the original implementation. This version is useful for multi-threaded programs.

There are NO guarantees about the quality of the pseudo random generator in standard Fortran. If you care about some particular quality of implementation for cryptography or science sensitive to random numbers (Monte-Carlo), you should use some library which you have control about.

You can study the manual of your compiler to find out what it says about the random number generator, but every compiler can implement a completely different algorithm to generate random numbers.

Numerical Recipes is actually not well received by some people in the numerical mathematics community http://www.uwyo.edu/buerkle/misc/wnotnr.html

This site is not for software recommendation, but this article (link given by roygvib in a comment): https://arxiv.org/abs/1005.4117 is a good review with examples of bad and good algorithms, methods how to test them, how to generate arbitrary number distributions and examples of calls of two example libraries in C (one of them can be called from Fortran as well).

Personally I use this https://bitbucket.org/LadaF/elmm/src/master/src/rng_par_zig.f90 parallel PRNG, but I didn't test the quality, I personally just need speed. But this is not a software recommendation site.