The actual "QuickBooks Bank" is through a partnership Intuit/QuickBooks has with Green Dot Bank. So if you already have a QBs Bank Acct, see if you can find a Green Dot Bank branch near you and confirm with them that you can come in to do QB Bank business.
If you're looking to open a Bank Acct that will work with QBs, you can go to any number of Banks that will work with Bank Feed and Online Payments. I use Chase, but whatever bank is close to you, just give them a call or drop in to talk to them.
DO check the Fees!! Some banks charge to do the Bank Feed and/or to Write Checks through QBs. Often this is based on the Balance you keep, but often it's just policy. I have no Fees to use Bank Feed, but I'm limited on sending out checks if I do it through QBs. So for the few checks I do write, I just go to the Banks Website and use their Online Payment system which is free. I funnel as many payments as I can through my Credit Card. (Just watch if some charge a CC Fee, those are the ones I send a check to.)