Try this pure-bash regex solution
shopt -s nocasematch #Dont care about the character case
text='<a href="hTTtp://www.google.com/search/something/lulz/here2;i=!mfo1iu489fn1o2jlk21m4098mdoi">"test link"</a><br>'
regex='(<a\ +href=\")([^\"]+)(\">)'
[[ $text =~ $regex ]] && echo ${BASH_REMATCH[2]}