Hep c and hep b are both blood borne viruses. There are also other blood borne viruses.
If there was a risk of hepatitis B or C infected blood entering the blood stream there could be a risk of transmission. This would be if they were using the same ink container on several customers instead of transferring to a smaller container and disposing of the excess material when done. Also the risk is much greater if they don’t sterilize the tattoo equipment between customers by using an autoclave.
From the hep b foundation:
“Unless you are vaccinated against hepatitis B, you are at risk for HBV if you are tattooed or pierced under unsterile conditions. ... That's thousands of tiny needle sticks per minute, and a very effective transmission route for blood borne pathogens like HBV, HCV and HIV.”
From the Mayo Clinic:
“Bloodborne diseases. If the equipment used to create your tattoo is contaminated with infected blood, you can contract various bloodborne diseases — including methicillin-resistant Staphylococcus aureus (MRSA), hepatitis B and hepatitis C. MRI complications.”
If concerned wait the appropriate amount of time and get tested. For hep c the wait is 12 weeks before having the hep c antibody test.
I believe if you think you were exposed to hep B if treated early you can prevent infection.
You should discuss your concerns with your doctor.