Try this:
q=$(echo $a | awk '{printf("\\\"%s\"",$1)}')
Test:
$ a=hi
$ echo $a
hi
$ q=$(echo $a | awk '{printf("\\\"%s\"",$1)}')
$ echo $q
\"hi"
Update:
It will, it just gets a littler messier.
q=`echo $a | awk '{printf("\\\\\"%s\"",$1)}'`
Test:
$ b=hello
$ echo $b
hello
$ t=`echo $b | awk '{printf("\\\\\"%s\"",$1)}'`
$ echo $t
\"hello"