I put my web expenses as a sub account of communication expenses since it's really a blog and I can communicate with clients through it. I also use communication expenses for things like VoIP since I don't have a land line phone.
The point is, it really doesn't matter where you put it. A deduction is a deduction. The point of the chart of accounts is to help you generate meaningful reports. When you get to tax time, if you put this deduction under office expenses or utilities rather than other expense, it really doesn't matter. If you got audited and you were a complete mess...auto deductions under meals, travel under office supplies, you give the IRS a reason to dig deeper, but when it comes down to it, an allowable expense is an allowable expense. Track the expenses in a way that best helps you run your business.
Is this a new company setup? Then you have many options; there is an account already created called "Computer and Internet Expense" (Acct #61700 if you have account numbers turned on in your Preferences) in many versions of QB.
If it is not a new company setup, be sure to check your preexisting expense accounts before creating a new account so you don't create a duplicate internet expense account. It could be confusing later on if you have been accruing expenses for the same service in two or more accounts.
If you do create a new account and you are a sole proprietorship using QB to help with tax returns, just be sure you put the tax-line mapping to Schedule C: Other business expenses so you get the business deduction come tax time.