Time to change your grammar checkers because the sentence is far from correct.
Clearly you've already realised that it would be more correct as.........
"Love is like life and death, don't know when it'll begin and when it'll end".
However, the sentence as it stands is a poorly expressed one. It would sound better as ........
"Love is like life and death: we know not when it will begin nor when it will end."
buggles (not a teacher)