Will pears ripen further after being picked?

Ripening
When you bring unripened pears home, store them in a bowl at room temperature. They should ripen within a week. Placing them in a paper bag can speed up the process. You can also add apples or bananas to the bag; this increases the concentration of fruit-ripening gases.
Check the pears daily. When the flesh around the stem feels slightly soft when pressed, the fruit is ready to eat. If you do not need the fruit yet at this point, you can move them to the refrigerator. This will slow down further ripening. It does not stop the process entirely; they will only be good in the refrigerator for another few days.

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Pears have to ripen off the tree. The texture is terrible if left to ripen before being picked.

GAs from both @thorninmud and @Trillian! Just to add one more thing: Bartletts are the only ones that turn yellow upon ripening. The other varieties don’t change color, so you have to do what @Trillian said, and you can’t depend on a color change.

All fruits continue to ripen after picking, before going bad. Vegetables just go bad with time.