Below is a shell script that replaces defined strings inside a group of files with the same extension. It is probably an overkill in your case since you can simply enter 4 mv commands instead of 2 needed with this script. First save the script as fix-name.com and make it executable (chmod +x fix-name.com). You also need to have a (t)csh installed, which I guess is not a given these days. I am sure someone will come up with a better bash script in no time.

In your case, enter:

fix-name.com fastq GEN191010_N_ ""
fix-name.com fastq _lib94257_1700 ""

The script:

#!/bin/tcsh
if ( "$1" == "" ) then
    echo ""
    echo " This script renames all files with a given extension by"
    echo " replacing part of their names with user specified strings."
    echo ""
    echo " The correct syntax is:"
    echo ""
    echo " fix-name.com <file extension> <replace what> <replace with>"
    echo ""
    echo " For example, to rename all *junk.txt files so that junk"
    echo " is removed from their names, use this command:"
    echo " "
    echo " fix-name.com txt junk ''"
    echo " "
    echo " First argument (file extension without .) has to be entered."
    echo " The defaults are junk and an empty string, which means"
    echo " removing junk from file names."
    echo ""
    exit 9
endif

if ( "$2" == "" ) then
    setenv STR1 "junk"
    else
    setenv STR1 $2
    endif

if ( "$3" == "" ) then
    setenv STR2 ""
    else
    setenv STR2 $3
    endif

find . -maxdepth 1 -name "*.$1" -print | agrep "$STR1" | sort > tmp-list1
cp tmp-list1 tmp-list2
perl -pi -e 's/\.\///g' tmp-list2
perl -pi -e 's/$ENV{"STR1"}/$ENV{"STR2"}/g' tmp-list2
perl -pi -e 's/\.\//mv /g' tmp-list1
paste -d" " tmp-list1 tmp-list2 > tmp-list
source tmp-list >& /dev/null
rm tmp-list tmp-list1 tmp-list2

Making assumptions about the consistency of your files:

for file in /path/to/*.fastq ; do 
    mv $file $(echo $file | sed -e 's/GEN191010_N_//gi'  -e 's/lib[0-9]\{5\}_[0-9]\{4\}_//gi')
done

NB: untested code.

An alternative (and the best) approach, using Unix's rename:

rename -nv 's/GEN191010_N_(.*)_lib[0-9]{5}_[0-9]{4}_/$1/gi' *.fastq

Drop the -n if you're happy with the substitutions, and it will actually perform the replacement.

GNU Parallel lets you harness the power of perl regular expressions:

parallel --dry-run mv {=Q($_)=} {=Q(s/GEN\d+_N_(\w+)_lib\d+_\d+_(\d+_R\d)/$1_$2/)=} ::: *.fastq

note: run it once with the --dry-run to make sure it does what you want, then run again without to do the deed.

This can be established using a for loop in bash script

open terminal from your directory containing fastq files

for i in *.fastq
do
    mv $i $(echo $i | awk '{split($1,a,/_/); print a[3]"_"a[5]"_"a[6]"_"a[7]}')
done