Is there an injection you get for toothache, from the emergency dept?

Temporary. The injection would be very short acting. See DDS fo solve problem.
ER tooth treatment. Typically the er department at a hospital does not give an injection for tooth pain. If there is an oral surgeon on call and it is determined that the tooth needs to be extracted immediately, then the procedure may be performed at that time.
Maybe. If the er department has on hand any of the typical local anesthetics used in dentistry, and the syringes and needles, you may get lucky and get an injection to help quell discomfort for the short term. Hovewer, most er medical personell are not sufficiently trained in the injection techniques for dental anesthesia.
Toothache. Most common cause is infection. ER injection will block pain until it wears off, but will not cure the infection. Please see Dentist, Endodontist root canal specialist, or specialist a Oral Surgeon.