Yes,quotes should start with a capital letter .The begining of the quote is the first letter of a sentence which should be capitalized.Examples of popular quotes:
“Be yourself; everyone else is already taken.” ― Oscar Wilde
“I'm selfish, impatient and a little insecure. I make mistakes, I am out of control and at times hard to handle. But if you can't handle me at my worst, then you sure as hell don't deserve me at my best.” ― Marilyn Monroe
“Two things are infinite: the universe and human stupidity; and I'm not sure about the universe.” ― Albert Einstein
“So many books, so little time.” ― Frank Zappa