This is for a family law case ie: Child Support
Divorce / Separation Lawyer
The simple answer is - yes. The slightly longer answer is that you and your attorney should have signed an engagement letter (a/k/a retainer agreement) which sets forth how you will be billed, and how much you are to pay, as well as you and your lawyers' other rights and responsibilities. Please contact your attorney to discuss this matter. You should not be charged to talking about billing.
Lawyers typically bill for the time they spend doing things related to the case, and they usually say this, plain as day, in the fee agreement. Driving to court is doing something related to the case. If the lawyer weren't appearing in court on your behalf, he wouldn't have driven there. So he charges you for his time, and it is perfectly proper for him to do so.
Not legal advice as I don't practice law in Illinois. It's just my two cents on the facts you describe in light of general principles of law. If you need legal advice, please consult a lawyer who holds Illinois licensure. That's not me.
Social Security Lawyers
In addition to the helpful answer offered by your first-responder, Ms. Goldstein, let me ask why you think an attorney shouldn't charge for travel time? For an attorney, time and expertise is the stock in trade. If your attorney has agreed to work on an hourly basis for a rate that takes expertise into account, why should the attorney not charge to travel to a distant court to appear on your behalf? Unless the attorney has a driver and can perform gainful work while riding to your hearing, that attorney is in your service while in transit.
Best wishes for an acceptable outcome, and please remember to designate a best answer.
This answer is offered as a public service for general information only and may not be relied upon as legal advice.