Many psalms in Psalms are by King David and his contemporaries. Many proverbs in Proverbs are by his son, King Solomon. Obviously, that makes them related in history.
ONE: Psalms is the context for Proverbs.
No one, as far as we know, wrote and disseminated proverbs in Israel before Solomon. Perhaps some proverbs can be gleaned from the earlier books of the Bible, but they weren’t written as a way of understanding wisdom.
Also, as far as we know, no one sang at the Tabernacle. The sacrifices were done in silence. Moses never set up a choir for the sanctuary, but David did (1 Chronicles 25). The generation preceding Solomon was trained in God’s songs.
When Solomon says that “the fear of YHWH is the beginning of wisdom” (Proverbs 1:7), he is not only quoting Psalm 111:10, but also making a statement that describes his own wisdom as a followup to his father’s work in organizing the priesthood (1 Chronicles 24) and bringing the Ark of the Covenant back to Jerusalem. Proverbs is mature insight for a people who have been trained by coming into God’s presence, being blessed by Him, and learning to sing His words.
Worship is the gateway to wisdom.
TWO: The “gateway” to the Psalms also leads to the Proverbs.
Psalm 1 and Psalm 2 are commonly acknowledged as the gateway to the rest of the Psalms. But they both naturally read as a gateway to Proverbs as well. Psalm 1 with its contrast between the righteous and the wicked and its exhortation to meditate on God’s instruction, and Psalm 2 with its recommendation to kings to “be wise,” introduce many of the themes in Proverbs.
THREE: The Proverbs trains us to speak in a way similar to the Psalms.
What do the Psalms mean? You can preach a sermon on a psalm, but that’s obviously not the primary way God wants us to use the Psalms. If God inspired congregational songs then he wants us to sing them.
Likewise, the Proverbs are meant to be memorized and recited. Often, the meaning of the Proverb is unclear in itself and requires delving into the straightforward ethical teaching of Scripture (the Ten Commandments, etc) in order to be understood. The Proverbs are about something more than communicating moral truths.
Toddlers learn to talk, not by having definitions explained, but by constant exposure to speech in life until they start talking themselves and grow more skilled at communicating like their parents. Definitions are a small part of what is involved in learning to talk. The Bible is a book and it contains information. But it is also a means by which God trains us how to speak–and therefore how to think and feel about life.
Some parts of the Bible are especially (not exclusively) intended for such training. Thus, Psalms and Proverbs are for the practice of worship and the practice of wisdom.