Use [ parameter expansion ]
string='Banana "hallo" Apple "hey"'
echo "$string"
Banana "hallo" Apple "hey"
string=${string//\"/\\\"} # Note both '\' need '"' need to be escaped.
echo "$string"
Banana \"hallo\" Apple \"hey\"
A lil explanation
${var/pattern/replacement}
replaces one occurrence of pattern in var with replacement.
${var//pattern/replacement}
replaces all occurrences of pattern in var with replacement.
If the pattern or replacement contains characters like " or / with special meaning in shell, they need to be escaped to let shell treat them as literals.