。
She told me I should quit smoking.
Literal
She [topic-は] me [to-に] no-smoking [should-すべきだ] [quotative-と] said.
禁煙 (lit. 'forbid-smoke') is both a noun ('no-smoking, abstaining from tobacco') and the basis of the verb 禁煙する ('to quit smoking, to abstain from smoking'). The sign 禁煙 marks no-smoking zones throughout Japan. ~べき is one of Japan's strongest 'should' patterns — moral, logical, or socially expected obligation, stronger than ~たほうがいい ('would be better to') but softer than ~なければならない ('must'). Note the form すべき rather than するべき: with する, both are grammatical, but すべき (using the classical attributive す) is preferred in writing and formal speech.