Slogans are a way of preaching to the choir, not a way of converting the opposition. If you want to encourage people to vote, then, you will need to have actual discussions with them and explain the virtues of voting and the vices of political apathy. But if you need something short and pithy, here are two great political quotes from the ancient Greek statesman Pericles:
“Just because you do not take an interest in politics doesn’t mean politics won’t take an interest in you.”
“We do not say that a man who takes no interest in politics is a man who minds his own business; we say that he has no business here at all.”