I'm on the Team plan and I ran out of quota, would upgrading to Business help me in this case?

No, upgrading to the Business plan won't grant you additional quota. By upgrading, you'll have access to all Business-level features but you'll keep your previous quota.

You’ll need to purchase additional events separately through either Pay-As-You-Go (on-demand) or reserved volume and stay in your Team plan. You can manage this by going to Settings (bottom left) → Usage & Billing → Subscription → Manage Plan.

Note: Only users with the Owner or Billing role can make changes to the subscription.