You cannot change the number of quests shown. If you have more than five quests active, you can control which five are shown by going to the Journal and marking the checkbox next to up to five quests that you want to focus on first. This marks them as priorities so they'll be shown in favor of any others.

(Note: the priority indicator is one-way only. You cannot choose to exclude a particular quest from the on-screen list except by prioritizing five other quests. If you have fewer than five priority-checked quests, the game will choose other unchecked quests to finish filling all available spots in the on-screen list of current quests.)