General quiet contemplation:
* Contemplate: To think about something seriously and for a long time.
* Meditate: To think deeply and quietly, often to relax or find peace.
* Ponder: To think about something carefully and thoroughly.
* Reflect: To think carefully about something that has happened.
* Muse: To think about something in a dreamy or thoughtful way.
More specific types of quiet thinking:
* Ruminate: To think deeply and repeatedly about something, often in a worried or anxious way.
* Introspect: To examine your own thoughts and feelings.
* Brood: To think about something in a gloomy or unhappy way.
* Mellow: To relax and think calmly.
Figurative options:
* Dream: To imagine or think about something pleasant.
* Daydream: To think about something pleasant while you are awake.
The best word for you will depend on the context and the specific kind of quiet thinking you want to describe.