I have a mountain dwarf wizard who has spent most of his life studying magic. He is an expert in magic items and arcane lore, and an intense book nerd. Problem is, I can't come up with a reason for him to leave his cozy library and set off adventuring that isn't super edgy or extreme. Any ideas are appreciated. 

Adventuring offers new spells, new magic items, and gold for costly spell components plus the opportunity to ethically put combat / dangerous spells into practice. Wayward fireballs and lightning bolts are not great when practicing in your library or out in the town risking innocents. On the road, fireballing bandits is not only great practice but cleans up crime and makes the road safer for others. 

Conjuration wizards may want to travel new places and discover or set up permanent teleport circles. Magical trade routes could be profitable. 

Books are limited both in scope of what they cover and by the perspective of the author. Sometimes actually experiencing things yourself provides far more knowledge - untainted by somebody else's view and opinions. 

Perhaps you want to explore the benefits of necromancy and see what good it can do - but without the misguided, narrow-minded, oppressions of the local townsfolk. Travelling out with a party who don't care about you raising zombies to use for labour or as guards, will be a great way to explore this avenue of magic.

Perhaps you are interested in inventing spells and need to go out into the field to find new, rare, components and reagents?

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Lots of reasons a Wizard may want to go out 'sploring.

Does he have a boss? Like a mentor or a higher up in an organization? If you’re starting in Tier 1, your Wiz is still in the Apprentice/low Journeyman stages of his career. In fact, the “Apprentice Wizard” is a “1st level spellcaster.” An apprentice has a Master/Mistress that teaches them. In the Renaissance, a master painter would likely have several journeyman artists that helped them and maybe dozens of apprentices. People payed masters lots of money to take their children on as apprentices. Keep in mind that an apprenticeship included education, food, lodging, and even clothing in some. A master would take on a young student and basically raise them, or people who could afford it became apprentices themselves sometimes too. So, your PC might be the only apprentice of a solitary Arcanist, or one of maybe dozens of students.

The apprentices would typically do the more mass-produceable stuff that would keep bringing money into the place, for wizards that would typically have been cheap, basic scrolls, potions, and other Common magic items. But there are no “ye olde magic shoppes” this edition so I really don’t understand how anybody supposedly does it? (I don’t have that problem, I have magic shops and it’ll isn’t a problem, I just keep it modest and make magic dear enough that common and minor uncommon items are still wondrous to not flood the market.)

Anyway, once an apprentice had worked up from copying scrolls and general go-for work clearing rats out of basements and picking up the shopping, to maybe making stuff like a little pick-me-up, or a little put-you-down and perhaps being sent to help more advanced students to deal with Kobolds or Goblins or something, they were further along.

Then they would move onto something a little fancier, like clothes that mend themselves, dirt repellent armor, stuff that’ll sell. Also, more important missions, like accompanying the master on business, and having other apprentices help you with the Kobolds for a change. When the master would ask what you wanted to make or which missions you wanted to pursue, that’s when the apprentice would start working under their own name instead of the master’s name. That’s when they finally become journeymen and can start charging, or getting paid by the master to continue working for them. In D&D that would be 5th level.

Perhaps his motivation is to finally graduate to journeyman and become their own Wizard. A 1st-level Wiz on their first “solo” adventure going out, partying up with a band of others, taking on a quest, and coming back 4th level or higher would certainly fit that bill.

it could be as simple as a family member gone missing. maybe a former wife

perhaps you've discovered something in history, something embarrassing to your clan, and you dare not trust it to an adventuring group so you've taken it upon yourself to research to redeem your clan.

Reading theory is wonderful, but there comes a time when you have to see magic actually put to real use in the world; adventuring provides opportunities to try spells for edge cases rather than just the standard uses suggested in classic texts.

After enough field work then a new book can be written about your experiences.

Options:

1. Get buy-in from your DM to give you leads for specific items to track.  Indiana Jones was an items specialist but what got him going was the initial persuit of a golden idol from a booby-trapped Peruvian temple.  The scriptwriters then provided Indy with a series of further leads to keep him hooked.  Can your DM do something like that?

2. Say your character is bored.  They've learned loads of lore and now want to test their knowledge.

3. Get a different character concept.  It may feel contrived to have a character that's motivated by study to then run off after potentially random adventure threads.  It can be as easy to start with a character that's motivated by adventure.   

4. Have a split character concept, perhaps your bookish wizard in a respected institution has a secret yearning to get out and about to an extent that might even be an embarrassment if it were known by peers.

I have a very similar character concept, but my Wizard (Abjurer) is a Mark of Warding Dwarf.

The motivation I gave to him was more problematic. My Dwarf was against the blind faith his fellow dwarven brother held upon the gods. Majority of his clan mates were Fighters, Paladins and Cleric and he decided to leave his mountain to prove that a dwarf could build his own destiny without bending his knees to Moradin.