Pueri pueri, pueri puerilia tractant - Children are children, (therefore) children do childish things
Pulvis et umbra sumus - We are dust and shadow. (Horace)
Purgamentum init, exit purgamentum - Garbage in, garbage out

Puris omnia pura

To the pure all things are pure

Puri sermonis amator - A lover of pure speech. (Terence)
Qua - In so far as
Quad nesciunt eos non interficiet - What they don't know won't kill them
Quandoquidem inter nos sanctissima divitiarum maiestas, esti funesta pecunia templo nondum habitas - Among us, the god most revered is Wealth, but so far it has no temple of its own
Quae nocent, saepe docent - What hurts, often instructs. One learns by bitter/adverse experience
Quae vide (qqv) - See these things