A quote may be used to convey a certain sentiment at any point in a valedictory speech based on the speaker's personal communication style and message. A quote may be most effective when used either in the introduction or in the conclusion.
When using a quote in the introduction, it is common to begin with a greeting for those to whom the speech is addressed. However, the use of a quote to begin the speech may create the tone that speaker wishes to achieve in motivating or inspiring peers. It is helpful to choose a short quote that is relevant to the occasion, making sure to give credit to the author.