You can use String.indexOf combined with String.substring:
void main() {
const str = "the quick brown fox jumps over the lazy dog";
const start = "quick";
const end = "over";
final startIndex = str.indexOf(start);
final endIndex = str.indexOf(end, startIndex + start.length);
print(str.substring(startIndex + start.length, endIndex)); // brown fox jumps
}
Note also that the startIndex is inclusive, while the endIndex is exclusive.